Displaying 1-10 of 39 results found.
Partial sums of powers of 16.
+10
52
1, 17, 273, 4369, 69905, 1118481, 17895697, 286331153, 4581298449, 73300775185, 1172812402961, 18764998447377, 300239975158033, 4803839602528529, 76861433640456465, 1229782938247303441, 19676527011956855057, 314824432191309680913, 5037190915060954894609
COMMENTS
16 = 2^4 is the growth measure for the Jacobsthal spiral (compare with phi^4 for the Fibonacci spiral). - Paul Barry, Mar 07 2008 Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=16, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 1, a(n-1) = det(A). - Milan Janjic, Feb 21 2010 a(n) is the total number of holes in a certain box fractal (start with 16 boxes, 1 hole) after n iterations. See illustration in links. - Kival Ngaokrajang, Jan 28 2015 Except for 1 and 17, all terms are Brazilian repunits numbers in base 16, and so belong to A125134. All terms >= 273 are composite because a(n) = ((4^(n+1) + 1) * (4^(n+1) - 1))/15. - Bernard Schott, Jun 06 2017 The sequence in binary is 1, 10001, 100010001, 1000100010001, 10001000100010001, ... cf. Plouffe link, A330135. - Frank Ellermann, Mar 05 2020
FORMULA
a(n) = if n=0 then 1 else a(n-1) + A001025(n).
EXAMPLE
a(3) = 1 + 16 + 256 + 4096 = 4369 = in binary: 1000100010001.
a(4) = (16^5 - 1)/15 = (4^5 + 1) * (4^5 - 1)/15 = 1025 * 1023/15 = 205 * 341 = 69905 = 11111_16. - Bernard Schott, Jun 06 2017
MATHEMATICA
Accumulate[16^Range[0, 20]] (* or *) LinearRecurrence[{17, -16}, {1, 17}, 20] (* Harvey P. Dale, Jul 19 2019 *)
PROG
(Sage) [gaussian_binomial(n, 1, 16) for n in range(1, 18)] # Zerinvary Lajos, May 28 2009 (Maxima)
a[0]:0$
a[n]:=16*a[n-1]+1$
(Python)
CROSSREFS
Cf. A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A091045, A218721, A218722, A064108, A218724- A218734, A132469, A218736- A218753, A133853, A094028, A218723. - M. F. Hasler, Nov 05 2012
Partial sums of powers of 17 ( A001026).
+10
41
1, 18, 307, 5220, 88741, 1508598, 25646167, 435984840, 7411742281, 125999618778, 2141993519227, 36413889826860, 619036127056621, 10523614159962558, 178901440719363487, 3041324492229179280, 51702516367896047761
COMMENTS
17^a(n) is largest power of 17 dividing (17^n)!.
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=17, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Feb 21 2010
FORMULA
a(n) = Sum_{k=0..n-1} 17^k = (17^n - 1)/16.
G.f.: x/((1 - 17*x)*(1 - x))= (1/(1 - 17*x) - 1/(1 - x))/16.
MAPLE
ListTools:-PartialSums([seq(17^k, k=0..30)]); # Robert Israel, Feb 18 2018
PROG
(Sage) [gaussian_binomial(n, 1, 17) for n in range(1, 18)] # Zerinvary Lajos, May 28 2009 (Maxima) makelist(sum(17^k, k, 0, n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */
CROSSREFS
Cf. similar sequences of the form (k^n-1)/(k-1) with k prime: A000225 (k=2), A003462 (k=3), A003463 (k=5), A023000 (k=7), A016123 (k=11), A091030 (k=13), this sequence (k=17), A218722 (k=19), A218726 (k=23), A218732 (k=29), A218734 (k=31), A218740 (k=37), A218744 (k=41), A218746 (k=43), A218750 (k=47).
0, 1, 21, 421, 8421, 168421, 3368421, 67368421, 1347368421, 26947368421, 538947368421, 10778947368421, 215578947368421, 4311578947368421, 86231578947368421, 1724631578947368421, 34492631578947368421, 689852631578947368421, 13797052631578947368421, 275941052631578947368421
COMMENTS
Partial sums of powers of 20 ( A009964), q-integers for q=20: diagonal k=1 in triangle A022184. For n >= 1, a(n) is the total number of holes in a certain box fractal (start with 20 boxes, 1 hole) after n iterations. See illustration in links. - Kival Ngaokrajang, Jan 28 2015
FORMULA
a(0)=0, a(1)=1, a(n) = 21*a(n-1) - 20*a(n-2). - Harvey P. Dale, Oct 04 2012
EXAMPLE
From N. J. A. Sloane, Nov 04 2014: Can also be obtained by writing powers of 2 in a staggered array and adding them (cf. A249604). For example, a(9) is: ..........1
.........2
........4
.......8
.....16
....32
...64
.128
256
-----------
26947368421
MAPLE
a:=n->sum(20^(n-j), j=0..n): seq(a(n), n=0..15); # Zerinvary Lajos, Feb 11 2007
MATHEMATICA
(20^Range[20]-1)/19 (* or *) NestList[20#+1&, 1, 20] (* Harvey P. Dale, Oct 04 2012 *)
PROG
(Sage) [gaussian_binomial(n, 1, 20) for n in range(1, 17)] # Zerinvary Lajos, May 29 2009 (PARI) for (n=0, 100, write("b064108.txt", n, " ", (20^n - 1)/19)) \\ Harry J. Smith, Sep 07 2009
CROSSREFS
Cf. A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218722, A064108, A218724, ..., A218733, ..., A218743, ..., A218752, A094028.
EXTENSIONS
Edited and extended to offset 0 by M. F. Hasler, Nov 04 2012
0, 1, 22, 463, 9724, 204205, 4288306, 90054427, 1891142968, 39714002329, 833994048910, 17513875027111, 367791375569332, 7723618886955973, 162195996626075434, 3406115929147584115, 71528434512099266416, 1502097124754084594737, 31544039619835776489478
COMMENTS
Partial sums of powers of 21 ( A009965); q-integers for q=21: diagonal k=1 in triangle A022185. For n >= 1, 4*a(n) is the total number of holes in a certain box fractal (start with 21 boxes, 4 holes) after n iterations. See illustration in links. - Kival Ngaokrajang, Jan 27 2015
PROG
(Magma) [n le 2 select n-1 else 22*Self(n-1) - 21*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012
CROSSREFS
Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218725- A218734, A132469, A218736- A218753, A133853, A094028, A218723.
0, 1, 32, 993, 30784, 954305, 29583456, 917087137, 28429701248, 881320738689, 27320942899360, 846949229880161, 26255426126284992, 813918209914834753, 25231464507359877344, 782175399728156197665, 24247437391572842127616, 751670559138758105956097
COMMENTS
Partial sums of powers of 31 ( A009975).
FORMULA
G.f.: x/((1 - x)*(1 - 31*x)).
a(n) = 32*a(n-1) - 31*a(n-2) for n > 1.
a(n) = floor(31^n/30). (End)
PROG
(PARI) a(n)=31^n\30
(Magma) [n le 2 select n-1 else 32*Self(n-1)-31*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012
CROSSREFS
Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724- A218733, A132469, A218736- A218753, A133853, A094028, A218723.
0, 1, 33, 1057, 33825, 1082401, 34636833, 1108378657, 35468117025, 1134979744801, 36319351833633, 1162219258676257, 37191016277640225, 1190112520884487201, 38083600668303590433, 1218675221385714893857, 38997607084342876603425, 1247923426698972051309601
REFERENCES
A. K. Devaraj, "Minimum Universal Exponent Generalisation of Fermat's Theorem", in ISSN #1550-3747, Proceedings of Hawaii Intl Conference on Statistics, Mathematics & Related Fields, 2004.
FORMULA
a(n) = (32^n - 1)/31 = floor(32^n/31) = Sum_{k=0..n} 32^k. - M. F. Hasler, Nov 05 2012
PROG
(Sage) [gaussian_binomial(5*n, 1, 2)/31 for n in range(1, 17)] # Zerinvary Lajos, May 28 2009 (Magma) [n le 2 select n-1 else 33*Self(n-1) - 32*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012
CROSSREFS
Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724- A218734, A132469, A218736- A218753, A133853, A094028, A218723.
EXTENSIONS
Edited and extended to offset 0 by M. F. Hasler, Nov 05 2012
0, 1, 19, 343, 6175, 111151, 2000719, 36012943, 648232975, 11668193551, 210027483919, 3780494710543, 68048904789775, 1224880286215951, 22047845151887119, 396861212733968143, 7143501829211426575, 128583032925805678351
COMMENTS
Partial sums of powers of 18 ( A001027), q-integers for q=18: diagonal k=1 in triangle A022182. Except for 0, 1 and 19, all terms are Brazilian repunits numbers in base 18, and so belong to A125134. From n = 3 to n = 8286, all terms are composite. See link "Generalized repunit primes". As explained in the extensions of A128164, a(25667) = (18^25667 - 1)/17 would be (is) the smallest prime in base 18. (End)
EXAMPLE
a(3) = (18^3 - 1)/17 = 343 = 7 * 49; a(6) = (18^6 - 1)/17 = 2000719 = 931 * 2149. - Bernard Schott, May 01 2017
MATHEMATICA
Join[{0}, Accumulate[18^Range[0, 20]]] (* Harvey P. Dale, Nov 08 2012 *)
PROG
(Magma) [n le 2 select n-1 else 19*Self(n-1)-18*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012
CROSSREFS
Cf. A000225, A001027, A002275, A002450, A002452, A003462, A003463, A003464, A014901, A014935, A016123, A016125, A022182, A023000, A023001, A064108, A091030, A091045, A094028, A125134, A128164, A131865, A135518, A135519, A218722, A218724, A218733, A218743, A218752.
0, 1, 50, 2451, 120100, 5884901, 288360150, 14129647351, 692352720200, 33925283289801, 1662338881200250, 81454605178812251, 3991275653761800300, 195572507034328214701, 9583052844682082520350, 469569589389422043497151, 23008909880081680131360400
COMMENTS
Partial sums of powers of 49 ( A087752).
MATHEMATICA
Join[{0}, Accumulate[49^Range[0, 20]]] (* Harvey P. Dale, Apr 14 2023 *)
PROG
(Magma) [n le 2 select n-1 else 50*Self(n-1) - 49*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 08 2012
CROSSREFS
Cf. A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724- A218734, A132469, A218736- A218752, A133853, A094028, A218723.
0, 1, 34, 1123, 37060, 1222981, 40358374, 1331826343, 43950269320, 1450358887561, 47861843289514, 1579440828553963, 52121547342280780, 1720011062295265741, 56760365055743769454, 1873092046839544391983, 61812037545704964935440, 2039797239008263842869521
COMMENTS
Partial sums of powers of 33 ( A009977).
FORMULA
G.f.: x/((1 - x)*(1 - 33*x)).
a(n) = 34*a(n-1) - 33*a(n-2).
a(n) = floor(33^n/32). (End)
PROG
(Magma) [n le 2 select n-1 else 34*Self(n-1)-33*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012
CROSSREFS
Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724- A218734, A132469, A218737- A218753, A133853, A094028, A218723.
Triangle read by rows: T(n,k) = value of the n-th repunit in base (k+1) representation, 1<=k<=n.
+10
15
1, 3, 4, 7, 13, 21, 15, 40, 85, 156, 31, 121, 341, 781, 1555, 63, 364, 1365, 3906, 9331, 19608, 127, 1093, 5461, 19531, 55987, 137257, 299593, 255, 3280, 21845, 97656, 335923, 960800, 2396745, 5380840, 511, 9841, 87381, 488281, 2015539, 6725601, 19173961, 48427561, 111111111
LINKS
Eric Weisstein's World of Mathematics, Repunit
FORMULA
T(n, k) = Sum_{i=0..n-1} (k+1)^i.
T(n+1, k) = (k+1)*T(n, k) + 1.
EXAMPLE
First 4 rows:
1: [1]_2
2: [11]_2 ........ [11]_3
3: [111]_2 ....... [111]_3 ....... [111]_4
4: [1111]_2 ...... [1111]_3 ...... [1111]_4 ...... [1111]_5
_
1: 1
2: 2+1 ........... 3+1
3: (2+1)*2+1 ..... (3+1)*3+1 ..... (4+1)*4+1
4: ((2+1)*2+1)*2+1 ((3+1)*3+1)*3+1 ((4+1)*4+1)*4+1 ((5+1)*5+1)*5+1.
MATHEMATICA
Table[((k+1)^n -1)/k, {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Aug 15 2022 *)
PROG
(Magma) [((k+1)^n -1)/k : k in [1..n], n in [1..12]]; // G. C. Greubel, Aug 15 2022 (SageMath)
def A125118(n, k): return ((k+1)^n -1)/k
CROSSREFS
This triangle shares some features with triangle A104878. This triangle is a portion of rectangle A055129. Each term of A110737 comes from the corresponding row of this triangle. Columns (adjusting offset as necessary): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724, A218725, A218726, A218727, A218728, A218729, A218730, A218731, A218732, A218733, A218734, A132469, A218736, A218737, A218738, A218739, A218740, A218741, A218742, A218743, A218744, A218745, A218746, A218747, A218748, A218749, A218750, A218751, A218753, A218752.
Search completed in 0.021 seconds
| 