A009965 Powers of 21.

1, 21, 441, 9261, 194481, 4084101, 85766121, 1801088541, 37822859361, 794280046581, 16679880978201, 350277500542221, 7355827511386641, 154472377739119461, 3243919932521508681, 68122318582951682301, 1430568690241985328321, 30041942495081691894741, 630880792396715529789561, 13248496640331026125580781, 278218429446951548637196401

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Same as Pisot sequences E(1, 21), L(1, 21), P(1, 21), T(1, 21). Essentially same as Pisot sequences E(21, 441), L(21, 441), P(21, 441), T(21, 441). 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 21-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

Links

• 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 (21).

Crossrefs

Row 10 of A329332.

Programs

Magma

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

Mathematica

21^Range[0,20] (* or *) NestList[21#&,1,20] (* Harvey P. Dale, Aug 31 2023 *)

Maxima

A009965(n):=21^n$
makelist(A009965(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI

a(n)=21^n \\ Charles R Greathouse IV, Nov 18 2011

Sage

[lucas_number1(n,21,0) for n in range(1, 17)] # Zerinvary Lajos, Apr 29 2009

Formula

For A009966..A009992 we have g.f.: 1/(1-qx), e.g.f.: exp(qx), with q = 21, 22, ..., 48. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001

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

G.f.: 22/G(0) where G(k) = 1 - 2*x*(k+1)/(1 - 1/(1 - 2*x*(k+1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 10 2013