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A009989 Powers of 45.

%I A009989 #40 Jul 10 2025 16:04:09
%S A009989 1,45,2025,91125,4100625,184528125,8303765625,373669453125,
%T A009989 16815125390625,756680642578125,34050628916015625,1532278301220703125,
%U A009989 68952523554931640625,3102863559971923828125,139628860198736572265625,6283298708943145751953125,282748441902441558837890625
%N A009989 Powers of 45.
%C A009989 Same as Pisot sequences E(1, 45), L(1, 45), P(1, 45), T(1, 45). Essentially same as Pisot sequences E(45, 2025), L(45, 2025), P(45, 2025), T(45, 2025). See A008776 for definitions of Pisot sequences.
%C A009989 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 45-colored compositions of n such that no adjacent parts have the same color. - _Milan Janjic_, Nov 17 2011
%H A009989 T. D. Noe, <a href="/A009989/b009989.txt">Table of n, a(n) for n = 0..100</a>
%H A009989 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H A009989 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (45).
%F A009989 G.f.: 1/(1-45*x). - _Philippe Deléham_, Nov 24 2008
%F A009989 a(n) = 45^n; a(n) = 45*a(n-1), a(0)=1. - _Vincenzo Librandi_, Nov 21 2010
%F A009989 From _Elmo R. Oliveira_, Jul 10 2025: (Start)
%F A009989 E.g.f.: exp(45*x).
%F A009989 a(n) = A000244(n)*A001024(n) = A000351(n)*A001019(n). (End)
%t A009989 45^Range[0,20] (* _Harvey P. Dale_, May 09 2012 *)
%o A009989 (Magma)[45^n: n in [0..20]]; // _Vincenzo Librandi_, Nov 21 2010
%o A009989 (PARI) a(n)=45^n \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A009989 Cf. A000244, A000351, A001019, A001024, A008776.
%K A009989 nonn,easy
%O A009989 0,2
%A A009989 _N. J. A. Sloane_