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%I A000566 M4358 N1826 #223 Sep 07 2024 15:43:31
%S A000566 0,1,7,18,34,55,81,112,148,189,235,286,342,403,469,540,616,697,783,
%T A000566 874,970,1071,1177,1288,1404,1525,1651,1782,1918,2059,2205,2356,2512,
%U A000566 2673,2839,3010,3186,3367,3553,3744,3940,4141,4347,4558,4774,4995,5221,5452,5688
%N A000566 Heptagonal numbers (or 7-gonal numbers): n*(5*n-3)/2.
%C A000566 Binomial transform of (0, 1, 5, 0, 0, 0, ...). Binomial transform is A084899. - _Paul Barry_, Jun 10 2003
%C A000566 Row sums of triangle A131413. - _Gary W. Adamson_, Jul 08 2007
%C A000566 Sequence starting (1, 7, 18, 34, ...) = binomial transform of (1, 6, 5, 0, 0, 0, ...). Also row sums of triangle A131896. - _Gary W. Adamson_, Jul 24 2007
%C A000566 Also the partial sums of A016861, a zero added in front; therefore a(n) = n (mod 5). - _R. J. Mathar_, Mar 19 2008
%C A000566 Also sequence found by reading the line from 0, in the direction 0, 7, ..., and the line from 1, in the direction 1, 18, ..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. These parallel lines are the semi-axes perpendicular to the main axis A195015 in the same spiral. - _Omar E. Pol_, Oct 14 2011
%C A000566 Also sequence found by reading the line from 0, in the direction 0, 7, ... and the parallel line from 1, in the direction 1, 18, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - _Omar E. Pol_, Jul 18 2012
%C A000566 Partial sums give A002413. - _Omar E. Pol_, Jan 12 2013
%C A000566 The n-th heptagonal number equals the sum of the n consecutive integers starting at 2*n-1; for example, 1, 3+4, 5+6+7, 7+8+9+10, etc. In general, the n-th (2k+1)-gonal number is the sum of the n consecutive integers starting at (k-1)*n - (k-2). When k = 1 and 2, this result generates the triangular numbers, A000217, and the pentagonal numbers, A000326, respectively. - _Charlie Marion_, Mar 02 2022
%D A000566 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
%D A000566 E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
%D A000566 Leonard 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. 2.
%D A000566 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000566 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000566 David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 123.
%H A000566 Daniel Mondot, <a href="/A000566/b000566.txt">Table of n, a(n) for n = 0..10000</a> (first 1000 terms by T. D. Noe)
%H A000566 S. Barbero, U. Cerruti, and N. Murru, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barbero/barbero5.html">Transforming Recurrent Sequences by Using the Binomial and Invert Operators</a>, J. Int. Seq. 13 (2010) # 10.7.7., section 4.4.
%H A000566 C. K. Cook and M. R. Bacon, <a href="https://www.fq.math.ca/Papers1/52-4/CookBacon4292014.pdf">Some polygonal number summation formulas</a>, Fib. Q., 52 (2014), 336-343.
%H A000566 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=341">Encyclopedia of Combinatorial Structures 341</a>.
%H A000566 Bir Kafle, Florian Luca, and Alain Togbé, <a href="https://doi.org/10.33039/ami.2020.09.002">Pentagonal and heptagonal repdigits</a>, Annales Mathematicae et Informaticae, pp. 137-145.
%H A000566 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A000566 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A000566 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polnum01.jpg">Illustration of initial terms of A000217, A000290, A000326, A000384, A000566, A000567</a>.
%H A000566 B. Srinivasa Rao, <a href="http://www.fq.math.ca/Papers1/43-3/paper43-3-1.pdf">Heptagonal Numbers in the Pell Sequence and Diophantine Equations 2x^2 = y^2(5y - 3)^2 ± 2</a>, Fib. Quarterly, 43 (2005), 194-201.
%H A000566 B. Srinivasa Rao, <a href="http://www.fq.math.ca/Papers1/43-3/paper43-3-1.pdf">Heptagonal numbers in the associated Pell sequence and Diophantine equations x^2(5x - 3)^2 = 8y^2 ± 4</a>, Fib. Quarterly, 43 (2005), 302-306.
%H A000566 Leo Tavares, <a href="/A000566/a000566.jpg">Illustration</a>.
%H A000566 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeptagonalNumber.html">Heptagonal Number</a>.
%H A000566 <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>.
%H A000566 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A000566 G.f.: x*(1 + 4*x)/(1 - x)^3. _Simon Plouffe_ in his 1992 dissertation.
%F A000566 a(n) = C(n, 1) + 5*C(n, 2). - _Paul Barry_, Jun 10 2003
%F A000566 a(n) = Sum_{k = 1..n} (4*n - 3*k). - _Paul Barry_, Sep 06 2005
%F A000566 a(n) = n + 5*A000217(n-1) - _Floor van Lamoen_, Oct 14 2005
%F A000566 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for a(0) = 0, a(1) = 1, a(2) = 7. -  _Jaume Oliver Lafont_, Dec 02 2008
%F A000566 a(n+1) = A153126(n) + n mod 2; a(2*n+1) = A033571(n); a(2*(n+1)) = A153127(n) + 1. - _Reinhard Zumkeller_, Dec 20 2008
%F A000566 40*a(n)+ 9 = A017354(n-1). - Ken Rosenbaum, Dec 02 2009.
%F A000566 a(n) = 2*a(n-1) - a(n-2) + 5, with a(0) = 0 and a(1) = 1. - _Mohamed Bouhamida_, May 05 2010
%F A000566 a(n) = A000217(n) + 4*A000217(n-1). - _Vincenzo Librandi_, Nov 20 2010
%F A000566 a(n) = a(n-1) + 5*n - 4, with a(0) = 0. - _Vincenzo Librandi_, Nov 20 2010
%F A000566 a(n) = A130520(5*n). - _Philippe Deléham_, Mar 26 2013
%F A000566 a(5*a(n) + 11*n + 1) = a(5*a(n) + 11*n) + a(5*n + 1). - _Vladimir Shevelev_, Jan 24 2014
%F A000566 Sum_{n>=1} 1/a(n) = sqrt(1 - 2/sqrt(5))*Pi/3 + 5*log(5)/6 - sqrt(5)*log((1 + sqrt(5))/2)/3 = 1.32277925312238885674944226131... . See A244639. - _Vaclav Kotesovec_, Apr 27 2016
%F A000566 E.g.f.: x*(2 + 5*x)*exp(x)/2. - _Ilya Gutkovskiy_, Aug 27 2016
%F A000566 From _Charlie Marion_, May 02 2017: (Start)
%F A000566 a(n+m) = a(n) + 5*n*m + a(m);
%F A000566 a(n-m) = a(n) - 5*n*m + a(m) + 3*m;
%F A000566 a(n) - a(m) = (5*(n + m) - 3)*(n - m)/2.
%F A000566 In general, let P(k,n) be the n-th k-gonal number. Then
%F A000566 P(k,n+m) = P(k,n) + (k - 2)*n*m + P(k,m);
%F A000566 P(k,n-m) = P(k,n) - (k - 2)*n*m + P(k,m) + (k - 4)*m;
%F A000566 P(k,n) - P(k,m) = ((k-2)*(n + m) + 4 - k)*(n - m)/2.
%F A000566 (End)
%F A000566 a(n) = A147875(-n) for all n in Z. - _Michael Somos_, Jan 25 2019
%F A000566 a(n) = A000217(n-1) + A000217(2*n-1). - _Charlie Marion_, Dec 19 2019
%F A000566 Product_{n>=2} (1 - 1/a(n)) = 5/7. - _Amiram Eldar_, Jan 21 2021
%F A000566 a(n) + a(n+1) = (2*n+1)^2 + n^2 - 2*n. In general, if we let P(k,n) = the n-th k-gonal number, then P(k^2-k+1,n)+ P(k^2-k+1,n+1) = ((k-1)*n+1)^2 + (k-2)*(n^2-2*n) = ((k-1)*n+1)^2 + (k-2)*A005563(n-2). When k = 2, this formula reduces to the well-known triangular number formula: T(n) + T(n+1) = (n+1)^2. - _Charlie Marion_, Jul 01 2021
%e A000566 G.f. = x + 7*x^2 + 18*x^3 + 34*x^4 + 55*x^5 + 81*x^6 + 112*x^7 + ... - _Michael Somos_, Jan 25 2019
%p A000566 A000566 := proc(n)
%p A000566         n*(5*n-3)/2 ;
%p A000566 end proc:
%p A000566 seq(A000566(n),n=0..30); # _R. J. Mathar_, Oct 02 2020
%t A000566 Table[n (5n - 3)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 1, 7}, 50] (* _Harvey P. Dale_, Oct 13 2011 *)
%t A000566 Join[{0},Accumulate[Range[1,315,5]]] (* _Harvey P. Dale_, Mar 26 2016 *)
%t A000566 (* For Mathematica 10.4+ *) Table[PolygonalNumber[RegularPolygon[7], n], {n, 0, 48}] (* _Arkadiusz Wesolowski_, Aug 27 2016 *)
%t A000566 PolygonalNumber[7,Range[0,50]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jan 23 2021 *)
%o A000566 (Magma) a000566:=func< n | n*(5*n-3) div 2 >; [ a000566(n): n in [0..50] ];
%o A000566 (PARI) a(n) = n * (5*n - 3) / 2
%o A000566 (Maxima) makelist(n*(5*n-3)/2, n, 0, 20); /* _Martin Ettl_, Dec 11 2012 */
%o A000566 (Haskell)
%o A000566 a000566 n = n * (5 * (n - 1) + 2) `div` 2
%o A000566 a000566_list = scanl (+) 0 a016861_list  -- _Reinhard Zumkeller_, Jun 16 2013
%o A000566 (Python) # Intended to compute the initial segment of the sequence, not isolated terms.
%o A000566 def aList():
%o A000566      x, y = 1, 1
%o A000566      yield 0
%o A000566      while True:
%o A000566          yield x
%o A000566          x, y = x + y + 5, y + 5
%o A000566 A000566 = aList()
%o A000566 print([next(A000566) for i in range(49)]) # _Peter Luschny_, Aug 04 2019
%o A000566 (Python) [n*(5*n-3)//2 for n in range(50)] # _Gennady Eremin_, Mar 24 2022
%Y A000566 Cf. A014637, A014640, A014773, A014792, A069099, A131413, A131896, A134483, A000384.
%Y A000566 a(n)= A093562(n+1, 2), (5, 1)-Pascal column.
%Y A000566 Cf. A006564, A147875, A244639.
%Y A000566 Cf. sequences listed in A254963.
%K A000566 nonn,easy,nice
%O A000566 0,3
%A A000566 _N. J. A. Sloane_
%E A000566 Partially edited by _Joerg Arndt_, Mar 11 2010

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