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A015445 Generalized Fibonacci numbers: a(n) = a(n-1) + 9*a(n-2). 18

%I

%S 1,1,10,19,109,280,1261,3781,15130,49159,185329,627760,2295721,

%T 7945561,28607050,100117099,357580549,1258634440,4476859381,

%U 15804569341,56096303770,198337427839,703204161769,2488241012320

%N Generalized Fibonacci numbers: a(n) = a(n-1) + 9*a(n-2).

%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>=2, 10*a(n-2) equals the number of 10-colored compositions of n with all parts >=2, such that no adjacent parts have the same color. - _Milan Janjic_, Nov 26 2011

%H Vincenzo Librandi, <a href="/A015445/b015445.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,9)

%F a(n) = (((1+sqrt(37))/2)^(n+1) - ((1-sqrt(37))/2)^(n+1))/sqrt(37).

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*9^k. - _Paul Barry_, Jul 20 2004

%F a(n) = Sum_{k=0..n} binomial((n+k)/2, (n-k)/2)*(1+(-1)^(n-k))*3^(n-k)/2}. - _Paul Barry_, Aug 28 2005

%F a(n) = Sum_{k=0..n} A109466(n,k)*(-9)^(n-k). - _Philippe Deléham_, Oct 26 2008

%F a(n) = (1/37)*(1/2+(1/2)*sqrt(37))^n*sqrt(37)-(1/37)*(1/2-(1/2)*sqrt(37))^n*sqrt(37). - _Paolo P. Lava_, Oct 01 2008 (May produce sequence with a different offset.)

%F G.f.: (-9*x-1)/(9*x^2+x-1) a(n)=(-703*(1/2-sqrt(37)/2)^n + 199*sqrt(37)*(1/2-sqrt(37)/2)^n-333*(1/2+sqrt(37)/2)^n + 171*sqrt(37)*(1/2+sqrt(37)/2)^n)/(74*(5*sqrt(37)-14)). - _Alexander R. Povolotsky_, Oct 13 2010

%F a(n) = Sum_{1<=k<=n+1, k odd} C(n+1,k)*37^((k-1)/2))/2^n. - _Vladimir Shevelev_, Feb 05 2014

%t CoefficientList[Series[(1+9*x)/(1-x-9*x^2), {x,0,50}], x] (* or *) LinearRecurrence[{1,9}, {1,1}, 50] (* _G. C. Greubel_, Apr 30 2017 *)

%o (Sage) [lucas_number1(n,1,-9) for n in xrange(1, 25)] # _Zerinvary Lajos_, Apr 22 2009

%o (MAGMA) [ n eq 1 select 1 else n eq 2 select 1 else Self(n-1)+9*Self(n-2): n in [1..30] ]; // _Vincenzo Librandi_, Aug 23 2011

%o (PARI) a(n)=([0,1; 9,1]^n*[1;1])[1,1] \\ _Charles R Greathouse IV_, Oct 03 2016

%Y Cf. A015443, A015442, A026595.

%K nonn,easy

%O 0,3

%A _Olivier Gérard_

%E Edited by _N. J. A. Sloane_, Oct 11 2010

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Last modified November 22 03:20 EST 2019. Contains 329383 sequences. (Running on oeis4.)