OFFSET
0,2
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
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).
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
EXAMPLE
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
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 *)
PROG
(Magma) [13^n: n in [0..100]]; // Vincenzo Librandi, Nov 21 2010
(PARI) first(n)=powers(13, n) \\ Charles R Greathouse IV, Jun 17 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from James A. Sellers, Sep 19 2000
STATUS
approved