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A009965 Powers of 21.

%I A009965 #52 Aug 31 2023 18:04:09
%S A009965 1,21,441,9261,194481,4084101,85766121,1801088541,37822859361,
%T A009965 794280046581,16679880978201,350277500542221,7355827511386641,
%U A009965 154472377739119461,3243919932521508681,68122318582951682301,1430568690241985328321,30041942495081691894741,630880792396715529789561,13248496640331026125580781,278218429446951548637196401
%N A009965 Powers of 21.
%C A009965 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 A009965 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 A009965 T. D. Noe, <a href="/A009965/b009965.txt">Table of n, a(n) for n = 0..100</a>
%H A009965 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A009965 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (21).
%F A009965 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 A009965 a(n) = 21^n; a(n) = 21*a(n-1), n > 0, a(0)=1. - _Vincenzo Librandi_, Nov 21 2010
%F A009965 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 A009965 21^Range[0,20] (* or *) NestList[21#&,1,20] (* _Harvey P. Dale_, Aug 31 2023 *)
%o A009965 (Sage) [lucas_number1(n,21,0) for n in range(1, 17)] # _Zerinvary Lajos_, Apr 29 2009
%o A009965 (Magma) [21^n: n in [0..100]] // _Vincenzo Librandi_, Nov 21 2010
%o A009965 (PARI) a(n)=21^n \\ _Charles R Greathouse IV_, Nov 18 2011
%o A009965 (Maxima) A009965(n):=21^n$
%o A009965 makelist(A009965(n),n,0,30); /* _Martin Ettl_, Nov 07 2012 */
%Y A009965 Row 10 of A329332.
%K A009965 nonn,easy
%O A009965 0,2
%A A009965 _N. J. A. Sloane_