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A001022 Powers of 13.
(Formerly M4914 N2107)
66
1, 13, 169, 2197, 28561, 371293, 4826809, 62748517, 815730721, 10604499373, 137858491849, 1792160394037, 23298085122481, 302875106592253, 3937376385699289, 51185893014090757, 665416609183179841, 8650415919381337933, 112455406951957393129, 1461920290375446110677, 19004963774880799438801 (list; graph; refs; listen; history; text; internal format)
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
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
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.
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 *)
PROG
(Magma) [13^n: n in [0..100]]; // Vincenzo Librandi, Nov 21 2010
(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
CROSSREFS
Cf. A001021.
Sequence in context: A045593 A045597 A045600 * A195945 A228389 A020533
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from James A. Sellers, Sep 19 2000
STATUS
approved

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Last modified March 3 20:15 EST 2024. Contains 370512 sequences. (Running on oeis4.)