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

0,2

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).

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

Mathematica

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

Prog

(Sage) [lucas_number1(n, 21, 0) for n in range(1, 17)] # Zerinvary Lajos, Apr 29 2009
(Magma) [21^n: n in [0..100]] // Vincenzo Librandi, Nov 21 2010
(PARI) a(n)=21^n \\ Charles R Greathouse IV, Nov 18 2011
(Maxima) A009965(n):=21^n$
makelist(A009965(n), n, 0, 30); /* Martin Ettl, Nov 07 2012 */

Crossrefs

Row 10 of A329332.

Sequence in context: A224009 A167255 A171295 * A285873 A189433 A041842

Adjacent sequences: A009962 A009963 A009964 * A009966 A009967 A009968

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved