Displaying 1-10 of 31 results found.
Numerators of the partial sums of the reciprocals of the positive numbers (k + 1)*(6*k + 5) = A049452(k+1).
+20
3
1, 27, 1487, 71207, 423323, 5021921, 208393341, 19767960169, 9496615779853, 112702096556215, 7360072449683999, 524616965933727859, 526363371877036219, 43813027890740553917, 781806518388353706041, 148866078528885256002173, 15064339628673236669081953, 538212602352090865654383697
COMMENTS
The corresponding denominators are given in A294965. For the general case V(m,r;n) = Sum_{k=0..n} 1/((k + 1)*(m*k + r)) = (1/(m - r))*Sum_{k=0..n} (m/(m*k + r) - 1/(k+1)), for r = 1, ..., m-1 and m = 2, 3, ..., and their limits see a comment in A294512. Here [m,r] = [6,5]. The limit of the series is V(6,5) = lim_{n -> oo} V(6,5;n) = . The value is (3/2)*log(3) + 2*log(2) - (1/2)*Pi*sqrt(3) = 0.3135137477... given in A294966.
REFERENCES
Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189 - 193.
FORMULA
a(n) = numerator(V(6,5;n)) with V(6,5;n) = Sum_{k=0..n} 1/((k + 1)*(6*k + 5)) = Sum_{k=0..n} 1/ A049452(k+1) = Sum_{k=0..n} (1/(k + 5/6) - 1/(k + 1)) = -Psi(5/6) + Psi(n+11/6) - (gamma + Psi(n+2)) with the digamma function Psi and the Euler-Mascheroni constant gamma = -Psi(1) from A001620.
EXAMPLE
The rationals V(6,5;n), n >= 0, begin: 1/5, 27/110, 1487/5610, 71207/258060, 423323/1496748, 5021921/17462060, 208393341/715944460, 19767960169/67298779240, 9496615779853/32101517697480, ...
V(6,5;10^6) = 0.313513577 (Maple, 10 digits) to be compared with the rounded ten digits 0.3135137478 obtained from V(6,5) given in A294966.
MAPLE
map(numer, ListTools:-PartialSums([seq(1/(k+1)/(6*k+5), k=0..20)])); # Robert Israel, Nov 29 2017
MATHEMATICA
Table[Numerator[Sum[1/((k+1)*(6*k+5)), {k, 0, n}]], {n, 0, 25}] (* G. C. Greubel, Aug 29 2018 *)
PROG
(PARI) a(n) = numerator(sum(k=0, n, 1/((k + 1)*(6*k + 5)))); \\ Michel Marcus, Nov 27 2017 (Magma) [Numerator((&+[1/((k+1)*(6*k+5)): k in [0..n]])): n in [0..25]]; // G. C. Greubel, Aug 29 2018
Denominators of the partial sums of the reciprocals of the numbers (k + 1)*(6*k + 5) = A049452(k+1).
+20
3
5, 110, 5610, 258060, 1496748, 17462060, 715944460, 67298779240, 32101517697480, 378797908830264, 24621864073967160, 1748152349251668360, 1748152349251668360, 145096644987888473880, 2582720280784414835064, 490716853349038818662160, 49562402188252920684878160
COMMENTS
The corresponding numerators are given in A294964. There details are found.
FORMULA
a(n) = denominator(V(6,5;n)) with V(6,5;n) = Sum_{k=0..n} 1/((k + 1)*(6*k + 5)) = Sum_{k=0..n} 1/ A049452(k+1) = Sum_{k=0..n} (1/(k + 5/6) - 1/(k + 1)).
EXAMPLE
For the rationals V(6,5;n) see A294964.
MAPLE
map(denom, ListTools:-PartialSums([seq(1/(k+1)/(6*k+5), k=0..20)])); # Robert Israel, Nov 29 2017
PROG
(PARI) a(n) = denominator(sum(k=0, n, 1/((k + 1)*(6*k + 5)))); \\ Michel Marcus, Nov 27 2017
Decimal expansion of the sum of the reciprocals of the numbers (k+1)*(6*k+5) = A049452(k+1) for k >= 0.
+20
2
3, 1, 3, 5, 1, 3, 7, 4, 7, 7, 7, 0, 7, 2, 8, 3, 8, 0, 0, 3, 6, 2, 1, 4, 7, 1, 1, 8, 3, 6, 9, 0, 8, 0, 9, 4, 6, 9, 6, 1, 3, 6, 7, 3, 3, 3, 1, 5, 5, 2, 3, 8, 2, 2, 4, 8, 8, 5, 7, 4, 1, 1, 6, 3, 6, 0, 8, 4, 3, 9, 1, 2, 0, 7, 7, 7, 7, 2, 0, 5, 5, 9, 9, 5, 9, 6, 2, 8, 0, 3, 8, 9, 5, 3, 4, 5, 2, 5, 4
COMMENTS
In the Koecher reference v_6(5) = (1/6)*(present value V(6,5)) = 0.05225229129512..., given on p. 192 as (1/4)*log(3) + (1/3)*log(2) - Pi/(4*sqrt(3)).
REFERENCES
Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189-193.
FORMULA
Sum_{k>=0} 1/((6*n + 5)*(n + 1)) =: V(6,5) = (3/2)*log(3) + 2*log(2) - (1/2)*Pi*sqrt(3) = -Psi(5/6) + Psi(1) with the digamma function Psi and Psi(1) = -gamma = A001620. Equals Sum_{k>=2} zeta(k)/6^(k-1). - Amiram Eldar, May 31 2021
EXAMPLE
0.313513747770728380036214711836908094696136733315523822488574116360843...
MATHEMATICA
RealDigits[-PolyGamma[0, 5/6] + PolyGamma[0, 1], 10, 100][[1]] (* G. C. Greubel, Sep 05 2018 *)
PROG
(PARI) default(realprecision, 100); (3/2)*log(3) + 2*log(2) - (1/2)*Pi*sqrt(3) \\ G. C. Greubel, Sep 05 2018 (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); (3/2)*Log(3) + 2*Log(2) - (1/2)*Pi(R)*Sqrt(3); // G. C. Greubel, Sep 05 2018
Pentagonal numbers: a(n) = n*(3*n-1)/2.
(Formerly M3818 N1562)
+10
540
0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151
COMMENTS
The average of the first n (n > 0) pentagonal numbers is the n-th triangular number. - Mario Catalani (mario.catalani(AT)unito.it), Apr 10 2003
a(n) is the sum of n integers starting from n, i.e., 1, 2 + 3, 3 + 4 + 5, 4 + 5 + 6 + 7, etc. - Jon Perry, Jan 15 2004 Partial sums of 1, 4, 7, 10, 13, 16, ... (1 mod 3), a(2k) = k(6k-1), a(2k-1) = (2k-1)(3k-2). - Jon Perry, Sep 10 2004 Starting with offset 1 = binomial transform of [1, 4, 3, 0, 0, 0, ...]. Also, A004736 * [1, 3, 3, 3, ...]. - Gary W. Adamson, Oct 25 2007 If Y is a 3-subset of an n-set X then, for n >= 4, a(n-3) is the number of 4-subsets of X having at least two elements in common with Y. - Milan Janjic, Nov 23 2007 Solutions to the duplication formula 2*a(n) = a(k) are given by the index pairs (n, k) = (5,7), (5577, 7887), (6435661, 9101399), etc. The indices are integer solutions to the pair of equations 2(6n-1)^2 = 1 + y^2, k = (1+y)/6, so these n can be generated from the subset of numbers [1+ A001653(i)]/6, any i, where these are integers, confined to the cases where the associated k=[1+ A002315(i)]/6 are also integers. - R. J. Mathar, Feb 01 2008 Even octagonal numbers divided by 8. - Omar E. Pol, Aug 18 2011 Sequence found by reading the line from 0, in the direction 0, 5, ... and the line from 1, in the direction 1, 12, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011 The hyper-Wiener index of the star-tree with n edges (see A196060, example). - Emeric Deutsch, Sep 30 2011 More generally the n-th k-gonal number is equal to n + (k-2)* A000217(n-1), n >= 1, k >= 3. In this case k = 5. - Omar E. Pol, Apr 06 2013 Note that both Euler's pentagonal theorem for the partition numbers and Euler's pentagonal theorem for the sum of divisors refer more exactly to the generalized pentagonal numbers, not this sequence. For more information see A001318, A175003, A238442. - Omar E. Pol, Mar 01 2014 The Fuss-Catalan numbers are Cat(d,k)= [1/(k*(d-1)+1)]*binomial(k*d,k) and enumerate the number of (d+1)-gon partitions of a (k*(d-1)+2)-gon (cf. Schuetz and Whieldon link). a(n)= Cat(n,3), so enumerates the number of (n+1)-gon partitions of a (3*(n-1)+2)-gon. Analogous sequences are A100157 (k=4) and A234043 (k=5). - Tom Copeland, Oct 05 2014 Binomial transform of (0, 1, 3, 0, 0, 0, ...) ( A169585 with offset 1) and second partial sum of (0, 1, 3, 3, 3, ...). - Gary W. Adamson, Oct 05 2015 For n > 0, a(n) is the number of compositions of n+8 into n parts avoiding parts 2 and 3. - Milan Janjic, Jan 07 2016 a(n) is also the number of edges in the Mycielskian of the complete graph K[n]. Indeed, K[n] has n vertices and n(n-1)/2 edges. Then its Mycielskian has n + 3n(n-1)/2 = n(3n-1)/2. See p. 205 of the West reference. - Emeric Deutsch, Nov 04 2016 Also the number of maximal cliques in the n-Andrásfai graph. - Eric W. Weisstein, Dec 01 2017 Coefficients in the hypergeometric series identity 1 - 5*(x - 1)/(2*x + 1) + 12*(x - 1)*(x - 2)/((2*x + 1)*(2*x + 2)) - 22*(x - 1)*(x - 2)*(x - 3)/((2*x + 1)*(2*x + 2)*(2*x + 3)) + ... = 0, valid for Re(x) > 1. Cf. A002412 and A002418. Column 2 of A103450. - Peter Bala, Mar 14 2019 A generalization of the Comment dated Apr 10 2003 follows. (k-3)* A000292(n-2) plus the average of the first n (2k-1)-gonal numbers is the n-th k-gonal number. - Charlie Marion, Nov 01 2020 a(n+1) is the number of Dyck paths of size (3,3n+1); i.e., the number of NE lattice paths from (0,0) to (3,3n+1) which stay above the line connecting these points. - Harry Richman, Jul 13 2021
REFERENCES
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pages 2 and 311.
Raymond Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; p. 129.
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 1.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 284.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 64.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
André Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhäuser, Boston, 1984; see p. 186.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 98-100.
Douglas B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001.
LINKS
Michel Waldschmidt, Continued fractions, Ecole de recherche CIMPA-Oujda, Théorie des Nombres et ses Applications, 18 - 29 mai 2015: Oujda (Maroc).
FORMULA
Product_{m > 0} (1 - q^m) = Sum_{k} (-1)^k*x^a(k). - Paul Barry, Jul 20 2003 G.f.: x*(1+2*x)/(1-x)^3.
E.g.f.: exp(x)*(x+3*x^2/2).
a(n) = n*(3*n-1)/2.
a(n) = binomial(3*n, 2)/3. - Paul Barry, Jul 20 2003 a(0) = 0, a(1) = 1; for n >= 2, a(n) = 2*a(n-1) - a(n-2) + 3. - Miklos Kristof, Mar 09 2005 a(n) = Sum_{k=1..n} (2*n - k). - Paul Barry, Aug 19 2005 a(n) = binomial(n+1, 2) + 2*binomial(n, 2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(0) = 0, a(1) = 1, a(2) = 5. - Jaume Oliver Lafont, Dec 02 2008 a(n) = a(n+9) - A016777(n)*9. (End) a(3*a(n) + 4*n + 1) = a(3*a(n) + 4*n) + a(3*n+1).
A generalization. Let {G_k(n)}_(n >= 0) be sequence of k-gonal numbers (k >= 3). Then the following identity holds: G_k((k-2)*G_k(n) + c(k-3)*n + 1) = G_k((k-2)*G_k(n) + c(k-3)*n) + G_k((k-2)*n + 1), where c = A000124. (End) Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(sqrt(3)*Pi - 6*log(2))/3 = 0.85501000622865446... - Ilya Gutkovskiy, Jul 28 2016 In general, let P(k,n) be the n-th k-gonal number. Then P(k,m+n) = P(k,m) + (k-2)mn + P(k,n). - Charlie Marion, Apr 16 2017 Product_{n>=2} (1 - 1/a(n)) = 3/5. - Amiram Eldar, Jan 21 2021 (n+1)*(a(n^2) + a(n^2+1) + ... + a(n^2+n)) = n*(a(n^2+n+1) + ... + a(n^2+2n)). - Charlie Marion, Apr 28 2024
EXAMPLE
Illustration of initial terms:
.
. o
. o o
. o o o o
. o o o o o o
. o o o o o o o o o
. o o o o o o o o o o o
. o o o o o o o o o o o o o
. o o o o o o o o o o o o o o
. o o o o o o o o o o o o o o o
.
. 1 5 12 22 35
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {0, 1, 5}, 61] (* Harvey P. Dale, Dec 27 2011 *) pentQ[n_] := IntegerQ[(1 + Sqrt[24 n + 1])/6]; pentQ[0] = True; Select[Range[0, 3200], pentQ@# &] (* Robert G. Wilson v, Mar 31 2014 *) Join[{0}, Accumulate[Range[1, 312, 3]]] (* Harvey P. Dale, Mar 26 2016 *) (* For Mathematica 10.4+ *) Table[PolygonalNumber[RegularPolygon[5], n], {n, 0, 46}] (* Arkadiusz Wesolowski, Aug 27 2016 *) CoefficientList[Series[x (-1 - 2 x)/(-1 + x)^3, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)
PROG
(PARI) a(n)=n*(3*n-1)/2
(PARI) vector(100, n, n--; binomial(3*n, 2)/3) \\ Altug Alkan, Oct 06 2015 (PARI) is_a000326 = my(s); n==0 || (issquare (24*n+1, &s) && s%6==5) \\ Hugo Pfoertner, Aug 03 2023 (Haskell)
(Python) # Intended to compute the initial segment of the sequence, not isolated terms.
def aList():
x, y = 1, 1
yield 0
while True:
yield x
x, y = x + y + 3, y + 3
CROSSREFS
The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542. Cf. A001318 (generalized pentagonal numbers), A049452, A033570, A010815, A034856, A051340, A004736, A033568, A049453, A002411 (partial sums), A033579. See A220083 for a list of numbers of the form n*P(s,n)-(n-1)*P(s,n-1), where P(s,n) is the n-th polygonal number with s sides. Cf. A240137: sum of n consecutive cubes starting from n^3. Cf. similar sequences listed in A022288.
Generalized pentagonal numbers: m*(3*m - 1)/2, m = 0, +-1, +-2, +-3, ....
(Formerly M1336 N0511)
+10
272
0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335
COMMENTS
"Conway's relation twixt the triangular and pentagonal numbers: Divide the triangular numbers by 3 (when you can exactly):
0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 ...
0 - 1 2 .- .5 .7 .- 12 15 .- 22 26 .- .35 .40 .- ..51 ...
.....-.-.....+..+.....-..-.....+..+......-...-.......+....
"and you get the pentagonal numbers in pairs, one of positive rank and the other negative.
"Append signs according as the pair have the same (+) or opposite (-) parity.
"Then Euler's pentagonal number theorem is easy to remember:
"p(n-0) - p(n-1) - p(n-2) + p(n-5) + p(n-7) - p(n-12) - p(n-15) ++-- = 0^n
where p(n) is the partition function, the left side terminates before the argument becomes negative and 0^n = 1 if n = 0 and = 0 if n > 0.
"E.g. p(0) = 1, p(7) = p(7-1) + p(7-2) - p(7-5) - p(7-7) + 0^7 = 11 + 7 - 2 - 1 + 0 = 15."
(End)
The sequence may be used in order to compute sigma(n), as described in Euler's article. - Thomas Baruchel, Nov 19 2003 Number of levels in the partitions of n + 1 with parts in {1,2}.
a(n) is the number of 3 X 3 matrices (symmetrical about each diagonal) M = {{a, b, c}, {b, d, b}, {c, b, a}} such that a + b + c = b + d + b = n + 2, a,b,c,d natural numbers; example: a(3) = 5 because (a,b,c,d) = (2,2,1,1), (1,2,2,1), (1,1,3,3), (3,1,1,3), (2,1,2,3). - Philippe Deléham, Apr 11 2007 Also numbers a(n) such that 24*a(n) + 1 = (6*m - 1)^2 are odd squares: 1, 25, 49, 121, 169, 289, 361, ..., m = 0, +-1, +-2, ... . - Zak Seidov, Mar 08 2008 This sequence contains all members of A000332 and all nonnegative members of A145919. For values of n such that n*(3*n - 1)/2 belongs to A000332, see A145919. (End) Starting with offset 1 can be considered the first in an infinite set generated from A026741. Refer to the array in A175005. - Gary W. Adamson, Apr 03 2010 A general formula for the generalized k-gonal numbers is given by n*((k - 2)*n - k + 4)/2, n=0, +-1, +-2, ..., k >= 5. - Omar E. Pol, Sep 15 2011 a(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and 2*w = 2*x + y. - Clark Kimberling, Jun 04 2012 Generalized k-gonal numbers are second k-gonal numbers and positive terms of k-gonal numbers interleaved, k >= 5. - Omar E. Pol, Aug 04 2012 a(n) is the sum of the largest parts of the partitions of n+1 into exactly 2 parts. - Wesley Ivan Hurt, Jan 26 2013 Conway's relation mentioned by R. K. Guy is a relation between triangular numbers and generalized pentagonal numbers, two sequences from different families, but as triangular numbers are also generalized hexagonal numbers in this case we have a relation between two sequences from the same family. - Omar E. Pol, Feb 01 2013 Start with the sequence of all 0's. Add n to each value of a(n) and the next n - 1 terms. The result is the generalized pentagonal numbers. - Wesley Ivan Hurt, Nov 03 2014 (6k + 1) | a(4k). (3k + 1) | a(4k+1). (3k + 2) | a(4k+2). (6k + 5) | a(4k+3). - Jon Perry, Nov 04 2014 Enge, Hart and Johansson proved: "Every generalised pentagonal number c >= 5 is the sum of a smaller one and twice a smaller one, that is, there are generalised pentagonal numbers a, b < c such that c = 2a + b." (see link theorem 5). - Peter Luschny, Aug 26 2016 The Enge, et al. result for c >= 5 also holds for c >= 2 if 0 is included as a generalized pentagonal number. That is, 2 = 2*1 + 0. - Michael Somos, Jun 02 2018 Suggestion for title, where n actually matches the list and b-file: "Generalized pentagonal numbers: k(n)*(3*k(n) - 1)/2, where k(n) = A001057(n) = [0, 1, -1, 2, -2, 3, -3, ...], n >= 0" - Daniel Forgues, Jun 09 2018 & Jun 12 2018 Generalized k-gonal numbers are the partial sums of the sequence formed by the multiples of (k - 4) and the odd numbers ( A005408) interleaved, with k >= 5. - Omar E. Pol, Jul 25 2018 The last digits form a symmetric cycle of length 40 [0, 1, 2, 5, ..., 5, 2, 1, 0], i.e., a(n) == a(n + 40) (mod 10) and a(n) == a(40*k - n - 1) (mod 10), 40*k > n. - Alejandro J. Becerra Jr., Aug 14 2018 Only 2, 5, and 7 are prime. All terms are of the form k*(k+1)/6, where 3 | k or 3 | k+1. For k > 6, the value divisible by 3 must have another factor d > 2, which will remain after the division by 6. - Eric Snyder, Jun 03 2022 8*a(n) is the product of two even numbers one of which is n + n mod 2. - Peter Luschny, Jul 15 2022 a(n) is the dot product of [1, 2, 3, ..., n] and repeat[1, 1/2]. a(5) = 12 = [1, 2, 3, 4, 5] dot [1, 1/2, 1, 1/2, 1] = [1 + 1 + 3 + 2 + 5]. - Gary W. Adamson, Dec 10 2022 Every nonnegative number is the sum of four terms of this sequence [S. Realis]. - N. J. A. Sloane, May 07 2023
REFERENCES
Enoch Haga, A strange sequence and a brilliant discovery, chapter 5 of Exploring prime numbers on your PC and the Internet, first revised ed., 2007 (and earlier ed.), pp. 53-70.
Ross Honsberger, Ingenuity in Mathematics, Random House, 1970, p. 117.
Donald E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, (to appear), section 7.2.1.4, equation (18).
Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, 2nd ed., Wiley, NY, 1966, p. 231.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Johannes W. Meijer, Euler's Ship on the Pentagonal Sea, pdf and jpg. S. Realis, Question 271, Nouv. Corresp. Math., 4 (1878) 27-29.
FORMULA
Euler: Product_{n>=1} (1 - x^n) = Sum_{n=-oo..oo} (-1)^n*x^(n*(3*n - 1)/2).
Euler transform of length-3 sequence [2, 2, -1]. - Michael Somos, Mar 24 2011 a(-1 - n) = a(n) for all n in Z. a(2*n) = A005449(n). a(2*n - 1) = A000326(n). - Michael Somos, Mar 24 2011. [The extension of the recurrence to negative indices satisfies the signature (1,2,-2,-1,1), but not the definition of the sequence m*(3*m -1)/2, because there is no m such that a(-1) = 0. - Klaus Purath, Jul 07 2021] a(n) = 3 + 2*a(n-2) - a(n-4). - Ant King, Aug 23 2011 Product_{k>0} (1 - x^k) = Sum_{k>=0} (-1)^k * x^a(k). - Michael Somos, Mar 24 2011 G.f.: x*(1 + x + x^2)/((1 + x)^2*(1 - x)^3).
Sequence consists of the pentagonal numbers ( A000326), followed by A000326(n) + n and then the next pentagonal number. - Jon Perry, Sep 11 2003 a(n) = Sum_{k=1..floor((n+1)/2)} (n - k + 1). - Paul Barry, Sep 07 2005 If n even a(n) = a(n-1) + n/2 and if n odd a(n) = a(n-1) + n, n >= 2. - Pierre CAMI, Dec 09 2007 a(n)-a(n-1) = A026741(n) and it follows that the difference between consecutive terms is equal to n if n is odd and to n/2 if n is even. Hence this is a self-generating sequence that can be simply constructed from knowledge of the first term alone. - Ant King, Sep 26 2011 a(n) = (1/2)*ceiling(n/2)*ceiling((3*n + 1)/2). - Mircea Merca, Jul 13 2012 a(n) = floor((n + 1)/2)*((n + 1) - (1/2)*floor((n + 1)/2) - 1/2). - Wesley Ivan Hurt, Jan 26 2013 a(n) = a(n+14)- A001651(n)*7. (End) G.f.: x*G(0), where G(k) = 1 + x*(3*k + 4)/(3*k + 2 - x*(3*k + 2)*(3*k^2 + 11*k + 10)/(x*(3*k^2 + 11*k + 10) + (k + 1)*(3*k + 4)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 16 2013 a(n) = Sum_{i=1..n} numerator(i/2) = Sum_{i=1..n} denominator(2/i). - Wesley Ivan Hurt, Feb 26 2017 Quadrisection. For r = 0,1,2,3: a(r + 4*k) = 6*k^2 + sqrt(24*a(r) + 1)*k + a(r), for k >= 1, with inputs (k = 0) {0,1,2,5}. These are the sequences A049453(k), A033570(k), A033568(k+1), A049452(k+1), for k >= 0, respectively. - Wolfdieter Lang, Feb 12 2021 a(n) = a(n-4) + sqrt(24*a(n-2) + 1), n >= 4. - Klaus Purath, Jul 07 2021 Sum_{n>=1} (-1)^(n+1)/a(n) = 6*(log(3)-1). - Amiram Eldar, Feb 28 2022 E.g.f.: (x*(7 + 3*x)*cosh(x) + (1 + 5*x + 3*x^2)*sinh(x))/8. - Stefano Spezia, Aug 01 2024
EXAMPLE
G.f. = x + 2*x^2 + 5*x^3 + 7*x^4 + 12*x^5 + 15*x^6 + 22*x^7 + 26*x^8 + 35*x^9 + ...
MAPLE
A001318 := -(1+z+z**2)/(z+1)**2/(z-1)**3; # Simon Plouffe in his 1992 dissertation; gives sequence without initial zero
MATHEMATICA
Table[n*(n+1)/6, {n, Select[Range[0, 100], Mod[#, 3] != 1 &]}]
Select[Accumulate[Range[0, 200]]/3, IntegerQ] (* Harvey P. Dale, Oct 12 2014 *) CoefficientList[Series[x (1 + x + x^2) / ((1 + x)^2 (1 - x)^3), {x, 0, 70}], x] (* Vincenzo Librandi, Nov 04 2014 *) LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 2, 5, 7}, 70] (* Harvey P. Dale, Jun 05 2017 *) a[ n_] := With[{m = Quotient[n + 1, 2]}, m (3 m + (-1)^n) / 2]; (* Michael Somos, Jun 02 2018 *)
PROG
(PARI) {a(n) = (3*n^2 + 2*n + (n%2) * (2*n + 1)) / 8}; /* Michael Somos, Mar 24 2011 */ (PARI) {a(n) = if( n<0, n = -1-n); polcoeff( x * (1 - x^3) / ((1 - x) * (1-x^2))^2 + x * O(x^n), n)}; /* Michael Somos, Mar 24 2011 */ (PARI) {a(n) = my(m = (n+1) \ 2); m * (3*m + (-1)^n) / 2}; /* Michael Somos, Jun 02 2018 */ (Sage)
@CachedFunction
if n == 0 : return 0
inc = n//2 if is_even(n) else n
(Magma) [(6*n^2 + 6*n + 1 - (2*n + 1)*(-1)^n)/16 : n in [0..50]]; // Wesley Ivan Hurt, Nov 03 2014 (Magma) [(3*n^2 + 2*n + (n mod 2) * (2*n + 1)) div 8: n in [0..70]]; // Vincenzo Librandi, Nov 04 2014 (Haskell)
a001318 n = a001318_list !! n
(GAP) a:=[0, 1, 2, 5];; for n in [5..60] do a[n]:=2*a[n-2]-a[n-4]+3; od; a; # Muniru A Asiru, Aug 16 2018 (Python)
def a(n):
p = n % 2
return (n + p)*(3*n + 2 - p) >> 3
CROSSREFS
Indices of nonzero terms of A010815, i.e., the (zero-based) indices of 1-bits of the infinite binary word to which the terms of A068052 converge. Sequences of generalized k-gonal numbers: this sequence (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
0, 2, 8, 18, 32, 50, 72, 98, 128, 162, 200, 242, 288, 338, 392, 450, 512, 578, 648, 722, 800, 882, 968, 1058, 1152, 1250, 1352, 1458, 1568, 1682, 1800, 1922, 2048, 2178, 2312, 2450, 2592, 2738, 2888, 3042, 3200, 3362, 3528, 3698, 3872, 4050, 4232, 4418
COMMENTS
"If each period in the periodic system ends in a rare gas ..., the number of elements in a period can be found from the ordinal number n of the period by the formula: L = ((2n+3+(-1)^n)^2)/8..." - Nature, Jun 09 1951; Nature 411 (Jun 07 2001), p. 648. This produces the present sequence doubled up.
Let z(1) = i = sqrt(-1), z(k+1) = 1/(z(k)+2i); then a(n) = (-1)*Imag(z(n+1))/Real(z(n+1)). - Benoit Cloitre, Aug 06 2002 Maximum number of electrons in an atomic shell with total quantum number n. Partial sums of A016825. - Jeremy Gardiner, Dec 19 2004 Arithmetic mean of triangular numbers in pairs: (1+3)/2, (6+10)/2, (15+21)/2, ... . - Amarnath Murthy, Aug 05 2005 These numbers form a pattern on the Ulam spiral similar to that of the triangular numbers. - G. Roda, Oct 20 2010
Integral areas of isosceles right triangles with rational legs (legs are 2n and triangles are nondegenerate for n > 0). - Rick L. Shepherd, Sep 29 2009 Number of stars when distributed as in the U.S.A. flag: n rows with n+1 stars and, between each pair of these, one row with n stars (i.e., n-1 of these), i.e., n*(n+1)+(n-1)*n = 2*n^2 = A001105(n). - César Eliud Lozada, Sep 17 2012 Apparently the number of Dyck paths with semilength n+3 and an odd number of peaks and the central peak having height n-3. - David Scambler, Apr 29 2013 Sum of the partition parts of 2n into exactly two parts. - Wesley Ivan Hurt, Jun 01 2013 Consider primitive Pythagorean triangles (a^2 + b^2 = c^2, gcd(a, b) = 1) with hypotenuse c ( A020882) and respective odd leg a ( A180620); sequence gives values c-a, sorted with duplicates removed. - K. G. Stier, Nov 04 2013 Number of roots in the root systems of type B_n and C_n (for n > 1). - Tom Edgar, Nov 05 2013 This sequence appears also as the first and second member of the quartet [a(n), a(n), p(n), p(n)] of the square of [n, n, n+1, n+1] in the Clifford algebra Cl_2 for n >= 0. p(n) = A046092(n). See an Oct 15 2014 comment on A147973 where also a reference is given. - Wolfdieter Lang, Oct 16 2014 a(n) are the only integers m where ( A000005(m) + A000203(m)) = (number of divisors of m + sum of divisors of m) is an odd number. - Richard R. Forberg, Jan 09 2015 a(n) represents the first term in a sum of consecutive integers running to a(n+1)-1 that equals (2n+1)^3. - Patrick J. McNab, Dec 24 2016 Also the number of 3-cycles in the (n+4)-triangular honeycomb obtuse knight graph. - Eric W. Weisstein, Jul 29 2017 Also the Wiener index of the n-cocktail party graph for n > 1. - Eric W. Weisstein, Sep 07 2017 Numbers represented as the palindrome 242 in number base B including B=2 (binary), 3 (ternary) and 4: 242(2)=18, 242(3)=32, 242(4)=50, ... 242(9)=200, 242(10)=242, ... - Ron Knott, Nov 14 2017 a(n) is the square of the hypotenuse of an isosceles right triangle whose sides are equal to n. - Thomas M. Green, Aug 20 2019 Apart from 0, integers such that the number of even divisors ( A183063) is odd. Proof: every n = 2^q * (2k+1), q, k >= 0, then 2*n^2 = 2^(2q+1) * (2k+1)^2; now, gcd(2, 2k+1) = 1, tau(2^(2q+1)) = 2q+2 and tau((2k+1)^2) = 2u+1 because (2k+1)^2 is square, so, tau(2*n^2) = (2q+2) * (2u+1).
The 2q+2 divisors of 2^(2q+1) are {1, 2, 2^2, 2^3, ..., 2^(2q+1)}, so 2^(2q+1) has 2q+1 even divisors {2^1, 2^2, 2^3, ..., 2^(2q+1)}.
Conclusion: these 2q+1 even divisors create with the 2u+1 odd divisors of (2k+1)^2 exactly (2q+1)*(2u+1) even divisors of 2*n^2, and (2q+1)*(2u+1) is odd. (End)
a(n) with n>0 are the numbers with period length 2 for Bulgarian and Mancala solitaire. - Paul Weisenhorn, Jan 29 2022 Number of points at L1 distance = 2 from any given point in Z^n. - Shel Kaphan, Feb 25 2023
REFERENCES
Arthur Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973.
Martin Gardner, The Colossal Book of Mathematics, Classic Puzzles, Paradoxes and Problems, Chapter 2 entitled "The Calculus of Finite Differences," W. W. Norton and Company, New York, 2001, pages 12-13.
L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 44.
Alain M. Robert, A Course in p-adic Analysis, Springer-Verlag, 2000, p. 213.
FORMULA
a(n) = (-1)^(n+1) * A053120(2*n, 2). G.f.: 2*x*(1+x)/(1-x)^3.
a(n) = A002378(n-1) + A002378(n). - Joerg M. Schuetze (joerg(AT)cyberheim.de), Mar 08 2010 [Corrected by Klaus Purath, Jun 18 2020] For n > 0, a(n) = 1/coefficient of x^2 in the Maclaurin expansion of 1/(cos(x)+n-1). - Francesco Daddi, Aug 04 2011 a(n) = Sum_{j=1..n} Sum_{i=1..n} ceiling((i+j-n+4)/3). - Wesley Ivan Hurt, Mar 12 2015 Product_{n>=1} (1 + 1/a(n)) = sqrt(2)*sinh(Pi/sqrt(2))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(2)*sin(Pi/sqrt(2))/Pi. (End)
EXAMPLE
a(3) = 18; since 2(3) = 6 has 3 partitions with exactly two parts: (5,1), (4,2), (3,3). Adding all the parts, we get: 1 + 2 + 3 + 3 + 4 + 5 = 18. - Wesley Ivan Hurt, Jun 01 2013
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {2, 8, 18}, {0, 20}] (* Eric W. Weisstein, Jul 28 2017 *)
CROSSREFS
Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488. Integers such that: this sequence (the number of even divisors is odd), A028982 (the number of odd divisors is odd), A028983 (the number of odd divisors is even), A183300 (the number of even divisors is even).
AUTHOR
Bernd.Walter(AT)frankfurt.netsurf.de
5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 167, 173, 179, 185, 191, 197, 203, 209, 215, 221, 227, 233, 239, 245, 251, 257, 263, 269, 275, 281, 287, 293, 299, 305, 311, 317, 323, 329, 335
COMMENTS
Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(18).
Exponents e such that x^e + x - 1 is reducible.
a(n-1), n >= 1, appears as first column in the triangle A239127 related to the Collatz problem. - Wolfdieter Lang, Mar 14 2014 Numbers that are not divisible by their digital root in base 4. - Amiram Eldar, Nov 24 2022
FORMULA
G.f.: (5+x)/(1-x)^2.
a(0) = 5; for n > 0, a(n) = a(n-1)+6.
(End)
a(n) = floor((12n-1)/2) with offset 1..a(1)=5. - Gary Detlefs, Mar 07 2010 a(n) = 3*Sum_{k = 0..n} binomial(6*n+5, 6*k+2)*Bernoulli(6*k+2). - Michel Marcus, Jan 11 2016 Sum_{n>=0} (-1)^n/a(n) = Pi/6 - sqrt(3)*arccoth(sqrt(3))/3. - Amiram Eldar, Dec 10 2021
MATHEMATICA
6Range[0, 59] + 5 (* or *) NestList[6 + # &, 5, 60] (* Harvey P. Dale, Mar 09 2013 *)
PROG
(Scala) (1 to 60).map(6 * _ - 1).mkString(", ") // Alonso del Arte, Nov 23 2018
Rhombic matchstick numbers: a(n) = n*(3*n+2).
+10
73
0, 5, 16, 33, 56, 85, 120, 161, 208, 261, 320, 385, 456, 533, 616, 705, 800, 901, 1008, 1121, 1240, 1365, 1496, 1633, 1776, 1925, 2080, 2241, 2408, 2581, 2760, 2945, 3136, 3333, 3536, 3745, 3960, 4181, 4408, 4641, 4880, 5125, 5376, 5633, 5896, 6165, 6440
COMMENTS
Write 1,2,3,4,... in a hexagonal spiral around 0, then a(n) is the n-th term of the sequence found by reading the line from 0 in the direction 0,5,.... The spiral begins:
.
85--84--83--82--81--80
. \
56--55--54--53--52 79
/ . \ \
57 33--32--31--30 51 78
/ / . \ \ \
58 34 16--15--14 29 50 77
/ / / . \ \ \ \
59 35 17 5---4 13 28 49 76
/ / / / . \ \ \ \ \
60 36 18 6 0 3 12 27 48 75
/ / / / / / / / / /
61 37 19 7 1---2 11 26 47 74
\ \ \ \ / / / /
62 38 20 8---9--10 25 46 73
\ \ \ / / /
63 39 21--22--23--24 45 72
\ \ / /
64 40--41--42--43--44 71
\ /
65--66--67--68--69--70
(End)
Connection to triangular numbers: a(n) = 4*T_n + S_n where T_n is the n-th triangular number and S_n is the n-th square. - William A. Tedeschi, Sep 12 2010 Sequence found by reading the line from 0, in the direction 0, 16, ... and the line from 5, in the direction 5, 33, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012 Let P denote the points from the n X n grid. A(n-1) also coincides with the minimum number of points Q needed to "block" P, that is, every line segment spanned by two points from P must contain one point from Q. - Manfred Scheucher, Aug 30 2018 Also the number of internal edges of an (n+1)*(n+1) "square" of hexagons; i.e., n+1 rows, each of n+1 edge-adjacent hexagons, stacked with minimal overhang. - Jon Hart, Sep 29 2019 For n >= 1, the continued fraction expansion of sqrt(27*a(n)) is [9n+2; {1, 2n-1, 1, 1, 1, 2n-1, 1, 18n+4}]. - Magus K. Chu, Oct 13 2022
FORMULA
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(0)=0, a(1)=5, a(2)=16. - Harvey P. Dale, May 06 2011 For n > 0, a(n)^3 + (a(n)+1)^3 + ... + (a(n)+n)^3 + 2* A000217(n)^2 = (a(n) + n + 1)^3 + ... + (a(n) + 2n)^3; see also A033954. - Charlie Marion, Dec 08 2007 Sum_{n>=1} 1/a(n) = (9 + sqrt(3)*Pi - 9*log(3))/12 = 0.3794906245574721941... . - Vaclav Kotesovec, Apr 27 2016 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/sqrt(12) - 3/4. - Amiram Eldar, Jul 03 2020
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {0, 5, 16}, 60] (* Harvey P. Dale, Jan 19 2016 *) CoefficientList[Series[x*(5 + x)/(1 - x)^3, {x, 0, 60}], x] (* Stefano Spezia, Sep 01 2018 *)
CROSSREFS
Cf. numbers of the form n*(d*n+10-d)/2: A008587, A056000, A028347, A140090, A014106, A028895, A186029, A007742, A022267, A033429, A022268, A049452, A186030, A135703, A152734, A139273.
0, 3, 14, 33, 60, 95, 138, 189, 248, 315, 390, 473, 564, 663, 770, 885, 1008, 1139, 1278, 1425, 1580, 1743, 1914, 2093, 2280, 2475, 2678, 2889, 3108, 3335, 3570, 3813, 4064, 4323, 4590, 4865, 5148, 5439, 5738, 6045, 6360, 6683, 7014, 7353, 7700, 8055, 8418
COMMENTS
Write 0,1,2,... in a clockwise spiral; sequence gives numbers on negative x axis. (See illustration in Example.)
This sequence is the number of expressions x generated for a given modulus n in finite arithmetic. For example, n=1 (modulus 1) generates 3 expressions: 0+0=0(mod 1), 0-0=0(mod 1), 0*0=0(mod 1). By subtracting n from 4n^2, we eliminate the counting of those expressions that would include division by zero, which would be, of course, undefined. - David Quentin Dauthier, Nov 04 2007 a(n) is also the Wiener index of the windmill graph D(3,n).
The windmill graph D(m,n) is the graph obtained by taking n copies of the complete graph K_m with a vertex in common (i.e., a bouquet of n pieces of K_m graphs). The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph.
Example: a(2)=14; indeed if the triangles are OAB and OCD, then, denoting distance by d, we have d(O,A)=d(O,B)=d(A,B)=d(O,C)=d(O,D)=d(C,D)=1 and d(A,C)=d(A,D)=d(B,C)=d(B,D)=2. The Wiener index of D(m,n) is (1/2)n(m-1)[(m-1)(2n-1)+1]. For the Wiener indices of D(4,n), D(5,n), and D(6,n) see A152743, A028994, and A180577, respectively. (End) Even hexagonal numbers divided by 2. - Omar E. Pol, Aug 18 2011 For n > 0, a(n) equals the number of length 3*n binary words having exactly two 0's with the n first bits having at most one 0. For example a(2) = 14. Words are 010111, 011011, 011101, 011110, 100111, 101011, 101101, 101110, 110011, 110101, 110110, 111001, 111010, 111100. - Franck Maminirina Ramaharo, Mar 09 2018 For n >= 1, the continued fraction expansion of sqrt(a(n)) is [2n-1; {1, 2, 1, 4n-2}]. For n=1, this collapses to [1; {1, 2}]. - Magus K. Chu, Sep 06 2022
REFERENCES
S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
FORMULA
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. - Harvey P. Dale, Oct 10 2011 E.g.f.: x*(3 + 4*x)*exp(x).
Sum_{n>=1} 1/a(n) = 3*log(2) - Pi/2 = 0.50864521488... (End)
a(n) = binomial(2*n, 2) + 2*n^2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi + log(3-2*sqrt(2)))/sqrt(2) - log(2). - Amiram Eldar, Mar 20 2022
EXAMPLE
Clockwise spiral (with sequence terms parenthesized) begins
16--17--18--19
|
15 4---5---6
| | |
(14) (3) (0) 7
| | | |
13 2---1 8
| |
12--11--10---9
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {0, 3, 14}, 50] (* Harvey P. Dale, Oct 10 2011 *)
CROSSREFS
Cf. A074378, A014848, A152743, A028994, A000326, A001105, A005476, A014635, A016742, A049452, A118729.
0, 5, 14, 27, 44, 65, 90, 119, 152, 189, 230, 275, 324, 377, 434, 495, 560, 629, 702, 779, 860, 945, 1034, 1127, 1224, 1325, 1430, 1539, 1652, 1769, 1890, 2015, 2144, 2277, 2414, 2555, 2700, 2849, 3002, 3159, 3320, 3485, 3654, 3827, 4004, 4185, 4370
COMMENTS
If Y is a 2-subset of a 2n-set X then, for n >= 1, a(n-1) is the number of (2n-2)-subsets of X intersecting Y. - Milan Janjic, Nov 18 2007 This sequence can also be derived from 1*(2+3)=5, 2*(3+4)=14, 3*(4+5)=27, and so forth. - J. M. Bergot, May 30 2011 Consider the partitions of 2n into exactly two parts. Then a(n) is the sum of all the parts in the partitions of 2n + the number of partitions of 2n + the total number of partition parts of 2n. - Wesley Ivan Hurt, Jul 02 2013 a(n) is the number of self-intersecting points of star polygon {(2*n+3)/(n+1)}. - Bui Quang Tuan, Mar 25 2015 a(n+1) is the number of function calls required to compute Ackermann's function ack(2,n). - Olivier Gérard, May 11 2018 a(n-1) is the least denominator d > n of the best rational approximation of sqrt(n^2-2) by x/d (see example and PARI code). - Hugo Pfoertner, Apr 30 2019 The number of cells in a loose n X n+1 rectangular spiral where n is even. See loose rectangular spiral image. - Jeff Bowermaster, Aug 05 2019
REFERENCES
Jolley, Summation of Series, Dover (1961).
FORMULA
G.f.: x*(5 - x)/(1 - x)^3. - Paul Barry, Feb 27 2003 Sum_{n>=1} 1/a(n) = 8/9 -2*log(2)/3 = 0.4267907685155920.. [Jolley eq. 265]
Sum_{n>=1} (-1)^(n+1)/a(n) = 4/9 + log(2)/3 - Pi/6. - Amiram Eldar, Jul 03 2020 a(n) = A000384(n+1) - 1. See Hex-tangles illustration. a(n) = A014105(n) + n*2. See Second Hex-tangles illustration. a(n) = 2* A002378(n) + n. See Ob-tangles illustration. a(n) = A005563(n) + 2* A000217(n). See Trap-tangles illustration. (End)
EXAMPLE
a(5-1) = 44: The best approximation of sqrt(5^2-2) = sqrt(23) by x/d with d <= k is 24/5 for all k < 44, but sqrt(23) ~= 211/44 is the first improvement. - Hugo Pfoertner, Apr 30 2019
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {0, 5, 14}, 50] (* Harvey P. Dale, Jul 21 2023 *)
PROG
(PARI) a(n)=2*n^2+3*n
(PARI) \\ least denominator > n in best rational approximation of sqrt(n^2-2)
for(n=2, 47, for(k=n, oo, my(m=denominator(bestappr(sqrt(n^2-2), k))); if(m>n, print1(k, ", "); break(1)))) \\ Hugo Pfoertner, Apr 30 2019
CROSSREFS
Cf. numbers of the form n*(d*n+10-d)/2: A008587, A056000, A028347, A140090, A028895, A045944, A186029, A007742, A022267, A033429, A022268, A049452, A186030, A135703, A152734, A139273.
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