A001022 Powers of 13.

1, 13, 169, 2197, 28561, 371293, 4826809, 62748517, 815730721, 10604499373, 137858491849, 1792160394037, 23298085122481, 302875106592253, 3937376385699289, 51185893014090757, 665416609183179841, 8650415919381337933, 112455406951957393129, 1461920290375446110677, 19004963774880799438801

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Same as Pisot sequences E(1, 13), L(1, 13), P(1, 13), T(1, 13). Essentially same as Pisot sequences E(13, 169), L(13, 169), P(13, 169), T(13, 169). 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 13-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

Numbers n such that sigma(13*n) = 13*n+sigma(n). - Jahangeer Kholdi, Nov 23 2013

Examples

			For the fifth formula: a(7) = 1*1 + 12*7 + 144*21 + 1728*35 + 20736*35 + 248832*21 + 2985984*7 + 35831808*1 = 62748517. - _Bruno Berselli_, Aug 06 2013
		

References

• 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

• T. D. Noe, Table of n, a(n) for n = 0..100
• P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 277
• Tanya Khovanova, Recursive Sequences
• Simon Plouffe, Approximations de Séries Génératrices et Quelques Conjectures, Dissertation, Université du Québec à Montréal, 1992.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
• Index entries for linear recurrences with constant coefficients, signature (13).

Crossrefs

Cf. A001021.

Programs

Magma

[13^n: n in [0..100]]; // Vincenzo Librandi, Nov 21 2010

Maple

A001022:=-1/(-1+13*z); # Simon Plouffe in his 1992 dissertation

Mathematica

Table[13^n, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)
13^Range[0,20] (* Harvey P. Dale, Jun 16 2024 *)

Maxima

A001022(n):=13^n$ makelist(A001022(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI

first(n)=powers(13,n) \\ Charles R Greathouse IV, Jun 17 2021

Formula

G.f.: 1/(1-13*x).

E.g.f.: exp(13*x).

a(n) = 13^n. - Vincenzo Librandi, Nov 21 2010

a(n) = 13*a(n-1) n > 0, a(0)=1. - Vincenzo Librandi, Nov 21 2010

a(n) = Sum_{k=0..n} A001021(k)*binomial(n,k). It is well known that Sum_{k=0..n} (h-1)^k*binomial(n,k) = h^n. - Bruno Berselli, Aug 06 2013

Extensions

More terms from James Sellers, Sep 19 2000