A001029 Powers of 19.

1, 19, 361, 6859, 130321, 2476099, 47045881, 893871739, 16983563041, 322687697779, 6131066257801, 116490258898219, 2213314919066161, 42052983462257059, 799006685782884121, 15181127029874798299, 288441413567621167681, 5480386857784802185939, 104127350297911241532841, 1978419655660313589123979, 37589973457545958193355601

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Same as Pisot sequences E(1, 19), L(1, 19), P(1, 19), T(1, 19). Essentially same as Pisot sequences E(19, 361), L(19, 361), P(19, 361), T(19, 361). 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 19-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

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 283
• Tanya Khovanova, Recursive Sequences
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• 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 (19).

Programs

Magma

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

Magma

[ n eq 1 select 1 else 19*Self(n-1): n in [1..21] ];

Maple

A001029:=-1/(-1+19*z); # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

Table[19^n,{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)

PARI

a(n)=19^n \\ Charles R Greathouse IV, Sep 24 2015

Formula

G.f.: 1/(1-19x), e.g.f.: exp(19x)

a(n) = 19^n; a(n) = 19*a(n-1) with a(0)=1. - Vincenzo Librandi, Nov 21 2010