%I #52 Aug 31 2023 18:04:09

%S 1,21,441,9261,194481,4084101,85766121,1801088541,37822859361,

%T 794280046581,16679880978201,350277500542221,7355827511386641,

%U 154472377739119461,3243919932521508681,68122318582951682301,1430568690241985328321,30041942495081691894741,630880792396715529789561,13248496640331026125580781,278218429446951548637196401

%N Powers of 21.

%C 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.

%C 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

%H T. D. Noe, <a href="/A009965/b009965.txt">Table of n, a(n) for n = 0..100</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (21).

%F 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

%F a(n) = 21^n; a(n) = 21*a(n-1), n > 0, a(0)=1. - _Vincenzo Librandi_, Nov 21 2010

%F 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

%t 21^Range[0,20] (* or *) NestList[21#&,1,20] (* _Harvey P. Dale_, Aug 31 2023 *)

%o (Sage) [lucas_number1(n,21,0) for n in range(1, 17)] # _Zerinvary Lajos_, Apr 29 2009

%o (Magma) [21^n: n in [0..100]] // _Vincenzo Librandi_, Nov 21 2010

%o (PARI) a(n)=21^n \\ _Charles R Greathouse IV_, Nov 18 2011

%o (Maxima) A009965(n):=21^n$

%o makelist(A009965(n),n,0,30); /* _Martin Ettl_, Nov 07 2012 */

%Y Row 10 of A329332.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_