A001024 -id:A001024 - cp's OEIS frontend

A133345 a(n) = 2a(n-1) + 14a(n-2) for n>1, a(0)=1, a(1)=1.

1, 1, 16, 46, 316, 1276, 6976, 31816, 161296, 768016, 3794176, 18340576, 89799616, 436367296, 2129929216, 10369000576, 50557010176, 246280028416, 1200358199296, 5848636796416, 28502288382976, 138885491915776

Offset: 0

Philippe Deléham, Dec 21 2007

Binomial transform of A001024 (powers of 15), with interpolated zeros.

a(n) is the number of compositions of n when there are 1 type of 1 and 15 types of other natural numbers. - Milan Janjic, Aug 13 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,14).

Cf. A001024, A098158.

[n le 2 select 1 else 2*(Self(n-1) +7*Self(n-2)): n in [1..41]]; // G. C. Greubel, Oct 15 2022

LinearRecurrence[{2,14},{1,1},30] (* Harvey P. Dale, Jan 07 2016 *)

Vec((1-x)/(1-2*x-14*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 12 2012

A133345=BinaryRecurrenceSequence(2,14,1,1)
[A133345(n) for n in range(41)] # G. C. Greubel, Oct 15 2022

G.f.: (1-x)/(1-2*x-14*x^2).
a(n) = Sum_{k=0..n} A098158(n,k)*15^(n-k). - Philippe Deléham, Dec 26 2007
If p[1]=1, and p[i]=15, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. - Milan Janjic, Apr 29 2010
a(n) = (b*i)^(n-1)*(b*i*ChebyshevU(n, -i/b) - ChebyshevU(n-1, -i/b)), with b = sqrt(14). - G. C. Greubel, Oct 15 2022

A009989 Powers of 45.

Original entry on oeis.org

1, 45, 2025, 91125, 4100625, 184528125, 8303765625, 373669453125, 16815125390625, 756680642578125, 34050628916015625, 1532278301220703125, 68952523554931640625, 3102863559971923828125, 139628860198736572265625, 6283298708943145751953125, 282748441902441558837890625

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 45), L(1, 45), P(1, 45), T(1, 45). Essentially same as Pisot sequences E(45, 2025), L(45, 2025), P(45, 2025), T(45, 2025). See A008776 for definitions of Pisot sequences.

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 45-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

T. D. Noe, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (45).

Cf. A000244, A000351, A001019, A001024, A008776.

[45^n: n in [0..20]]; // Vincenzo Librandi, Nov 21 2010

45^Range[0,20] (* Harvey P. Dale, May 09 2012 *)

a(n)=45^n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 1/(1-45*x). - Philippe Deléham, Nov 24 2008
a(n) = 45^n; a(n) = 45*a(n-1), a(0)=1. - Vincenzo Librandi, Nov 21 2010
From Elmo R. Oliveira, Jul 10 2025: (Start)
E.g.f.: exp(45*x).
a(n) = A000244(n)*A001024(n) = A000351(n)*A001019(n). (End)

A067420 Fifth column of triangle A067417.

Original entry on oeis.org

1, 7, 105, 1575, 23625, 354375, 5315625, 79734375, 1196015625, 17940234375, 269103515625, 4036552734375, 60548291015625, 908224365234375, 13623365478515625, 204350482177734375, 3065257232666015625

Offset: 0

Wolfdieter Lang, Jan 25 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..800
Index entries for linear recurrences with constant coefficients, signature (15).

Cf. A067419 (fourth column), A067421 (sixth column), A001024 (powers of 15).

[Ceiling(7*(3*5)^(n-1)): n in [0..20]]; // Vincenzo Librandi, Oct 02 2011

a(n) = A067417(n+4, 4).
a(n) = 7*(3*5)^(n-1), n >= 1, a(0)=1.
G.f.: (1-8*x)/(1-15*x).

A147716 Triangle of coefficients in expansion of (14 + x)^n.

Original entry on oeis.org

1, 14, 1, 196, 28, 1, 2744, 588, 42, 1, 38416, 10976, 1176, 56, 1, 537824, 192080, 27440, 1960, 70, 1, 7529536, 3226944, 576240, 54880, 2940, 84, 1, 105413504, 52706752, 11294304, 1344560, 96040, 4116, 98, 1, 1475789056, 843308032, 210827008, 30118144, 2689120, 153664, 5488, 112, 1

Offset: 0

Philippe Deléham, Nov 11 2008

Triangle T(n,k), read by rows, given by [14, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

			Triangle begins :
       1;
      14,      1;
     196,     28,     1;
    2744,    588,    42,    1;
   38416,  10976,  1176,   56,  1;
  537824, 192080, 27440, 1960, 70, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A001024, A023531, A130595.

Sequences of the form q^(n-k)*binomial(n, k): A007318 (q=1), A038207 (q=2), A027465 (q=3), A038231 (q=4), A038243 (q=5), A038255 (q=6), A027466 (q=7), A038279 (q=8), A038291 (q=9), A038303 (q=10), A038315 (q=11), A038327 (q=12), A133371 (q=13), this sequence (q=14), A027467 (q=15).

[14^(n-k)*Binomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 15 2021

With[{m=8}, CoefficientList[CoefficientList[Series[1/(1-14*x-x*y), {x, 0, m}, {y, 0, m}], x], y]]//Flatten (* Georg Fischer, Feb 17 2020 *)

flatten([[14^(n-k)*binomial(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 15 2021

T(n,k) = binomial(n,k) * 14^(n-k).
G.f.: 1/(1 - 14*x - x*y). - R. J. Mathar, Aug 12 2015
Sum_{k=0..n} T(n, k) = 15^n = A001024(n). - G. C. Greubel, May 15 2021

a(36) corrected by Georg Fischer, Feb 17 2020

A100403 Digital root of 6^n.

Original entry on oeis.org

1, 6, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 0

Cino Hilliard, Dec 31 2004

Also the digital root of k^n for any k == 6 (mod 9). - Timothy L. Tiffin, Dec 02 2023

			For n=8, the digits of 6^8 = 1679616 sum to 36, whose digits sum to 9. So, a(8) = 9. - _Timothy L. Tiffin_, Dec 01 2023

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000400, A001024, A007953, A009968, A009977, A009986, A010888, A100401, A159991.

PadRight[{1, 6}, 100, 9] (* Timothy L. Tiffin, Dec 03 2023 *)

a(n) = if( n<2, [1,6][n+1], 9); \\ Joerg Arndt, Dec 03 2023

From Timothy L. Tiffin, Dec 01 2023: (Start)
a(n) = 9 for n >= 2.
G.f.: (1+5x+3x^2)/(1-x).
a(n) = A100401(n) for n <> 1.
a(n) = A010888(A000400(n)) = A010888(A001024(n)) = A010888(A009968(n)) = A010888(A009977(n)) = A010888(A009986(n)) = A010888(A159991(n)). (End)
E.g.f.: 9*exp(x) - 3*x - 8. - Elmo R. Oliveira, Aug 09 2024
a(n) = A007953(6*a(n-1)) = A010888(6*a(n-1)). - Stefano Spezia, Mar 20 2025

A013720 a(n) = 15^(2*n + 1).

Original entry on oeis.org

15, 3375, 759375, 170859375, 38443359375, 8649755859375, 1946195068359375, 437893890380859375, 98526125335693359375, 22168378200531005859375, 4987885095119476318359375, 1122274146401882171630859375

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (225).

Bisection of A001024 (15^n).

[15^(2*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 26 2011

seq(15^(2*n+1),n=0..11); # Nathaniel Johnston, Jun 25 2011

15^(2Range[0,12]+1)  (* Harvey P. Dale, Feb 09 2011 *)

a(n)=15^(2*n+1) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 25 2008: (Start)
a(n) = 225*a(n-1), a(0)=15.
G.f.: 15/(1-225*x). (End)

A060220 Number of orbits of length n under the full 17-shift (whose periodic points are counted by A001026).

Original entry on oeis.org

17, 136, 1632, 20808, 283968, 4022064, 58619808, 871959240, 13176430176, 201599248032, 3115626937056, 48551851084080, 761890617915840, 12026987582075856, 190828203433892736, 3041324491793194440, 48661191875666868480, 781282469552728498992, 12582759772902701307744

Offset: 1

Thomas Ward, Mar 21 2001

Number of monic irreducible polynomials of degree n over GF(17). - Andrew Howroyd, Dec 10 2017

			a(2)=136 since there are 289 points of period 2 in the full 17-shift and 17 fixed points, so there must be (289-17)/2 = 136 orbits of length 2.

G. C. Greubel, Table of n, a(n) for n = 1..810
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
T. Ward, Exactly realizable sequences

Column 17 of A074650.

Cf. A001026, A008683.

A060220:= func< n | (1/n)*(&+[MoebiusMu(d)*(17)^Floor(n/d): d in Divisors(n)]) >;
[A060220(n): n in [1..40]]; // G. C. Greubel, Sep 13 2024

A060220[n_]:= DivisorSum[n, (17)^(n/#)*MoebiusMu[#] &]/n;
Table[A060220[n], {n,40}] (* G. C. Greubel, Sep 13 2024 *)

a001024(n) = 17^n;
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001024(n/d)); \\ Michel Marcus, Sep 11 2017

def A060220(n): return (1/n)*sum(moebius(k)*(17)^(n/k) for k in (1..n) if (k).divides(n))
[A060220(n) for n in range(1,41)] # G. C. Greubel, Sep 13 2024

a(n) = (1/n)* Sum_{d|n} mu(d)*A001026(n/d).

G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 17*x^k))/k. - Ilya Gutkovskiy, May 20 2019

More terms from Michel Marcus, Sep 11 2017

A239284 a(n) = (15^n - (-1)^n)/16.

Original entry on oeis.org

0, 1, 14, 211, 3164, 47461, 711914, 10678711, 160180664, 2402709961, 36040649414, 540609741211, 8109146118164, 121637191772461, 1824557876586914, 27368368148803711, 410525522232055664, 6157882833480834961, 92368242502212524414, 1385523637533187866211

Offset: 0

Felix P. Muga II, Mar 14 2014

Let k and t be positive integers and consider a(n) = k*a(n-1)+t*a(n-2) for n>=2, with a(0)=0, a(1)=1.
The roots of its characteristic equation are r1 = (k+sqrt(k^2+4t))/2 and r2 =(k-sqrt(k^2+4t))/2. Hence, the solution to the recurrence relation is the sequence {a(n)} where a(n) = alpha1*r1^n + alpha2*r2^n. It can be shown that alpha1 = 1/sqrt(k^2+4t) and alpha2 = -alpha1. It can be shown also that |r2/r1|< 1. Thus, the ratio a(n+1)/a(n) converges to r as n approaches infinity.
Note that limit a(n+1)/a(n) = 15 as n approaches infinity with k=14 and t=15.
If n > 15 then | a(n+1)/a(n) - 15 | < 10^(-16).
The number of walks of length n between any two distinct vertices of the complete graph K_16. - Peter Bala, May 30 2024

G. C. Greubel, Table of n, a(n) for n = 0..849
Index entries for linear recurrences with constant coefficients, signature (14,15).

Cf. A062160 (row 15).

[(15^n - (-1)^n)/16: n in [0..30]]; // G. C. Greubel, May 26 2018

CoefficientList[Series[x/(1-14*x-15*x^2), {x,0,50}], x] (* or *) Table[ (15^n - (-1)^n)/16, {n,0,30}] (* or *) LinearRecurrence[{14,15}, {0,1}, 30] (* G. C. Greubel, May 26 2018 *)

a(n) = (15^n - (-1)^n)/16; \\ Michel Marcus, Mar 16 2014

my(x='x+O('x^30)); concat([0], Vec(x/(1 -14*x - 15*x^2))) \\ G. C. Greubel, May 26 2018

G.f.: x/(1 - 14*x - 15*x^2).
a(n) = 14*a(n-1) + 15*a(n-2) for n > 1, a(0) = 0, a(1) = 1.
a(n) = (1/16)*(15^n - (-1)^n).
a(n) = (1/16)*( A001024(n) - A033999(n) ).
E.g.f.: (exp(15*x) - exp(-x))/16. - G. C. Greubel, May 26 2018

A013756 a(n) = 15^(3*n + 1).

Original entry on oeis.org

15, 50625, 170859375, 576650390625, 1946195068359375, 6568408355712890625, 22168378200531005859375, 74818276426792144775390625, 252511682940423488616943359375, 852226929923929274082183837890625, 2876265888493261300027370452880859375

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3375).

Subsequence of A001024.

[15^(3*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 27 2011

Table[15^(3n+1),{n,0,10}] (* or *) NestList[3375#&,15,10] (* Harvey P. Dale, Sep 06 2023 *)

a(n)=15^(3*n+1) \\ Charles R Greathouse IV, Jul 10 2016

From Philippe Deléham, Dec 03 2008: (Start)
a(n) = 3375*a(n-1); a(0)=15
G.f.: 15/(1-3375*x). (End)

A013757 a(n) = 15^(3*n + 2).

Original entry on oeis.org

225, 759375, 2562890625, 8649755859375, 29192926025390625, 98526125335693359375, 332525673007965087890625, 1122274146401882171630859375, 3787675244106352329254150390625, 12783403948858939111232757568359375

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3375).

Subsequence of A001024.

[15^(3*n+2): n in [0..15]]; // Vincenzo Librandi, Jun 27 2011

15^(3Range[0,20]+2) (* or *) NestList[3375#&,225,20] (* Harvey P. Dale, Oct 25 2011 *)

makelist(15^(3*n+2),n,0,20); /* Martin Ettl, Oct 21 2012 */

a(n)=15^(3*n+2) \\ Charles R Greathouse IV, Jul 10 2016

From Philippe Deléham, Dec 03 2008: (Start)
a(n) = 3375*a(n-1); a(0)=225.
G.f.: 225/(1-3375*x). (End)

cp's OEIS Frontend

A133345 a(n) = 2*a(n-1) + 14*a(n-2) for n>1, a(0)=1, a(1)=1.

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A009989 Powers of 45.

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A067420 Fifth column of triangle A067417.

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A147716 Triangle of coefficients in expansion of (14 + x)^n.

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A100403 Digital root of 6^n.

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A013720 a(n) = 15^(2*n + 1).

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A060220 Number of orbits of length n under the full 17-shift (whose periodic points are counted by A001026).

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A133345 a(n) = 2a(n-1) + 14a(n-2) for n>1, a(0)=1, a(1)=1.