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%I #57 Dec 30 2023 23:40:18
%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 J. Borowska, L. Lacinska, <a href="https://doi.org/10.17512/jamcm.2014.4.03">Recurrence form of determinant of a heptadiagonal symmetric Toeplitz matrix</a>, J. Appl. Math. Comp. Mech. 13 (2014) 19-16, remark 2 for permanent of tridiagonal Toeplitz matrices a=1, b=3.
%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) = (-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
%F G.f.: 1/(1-x-9*x^2). - _Philippe Deléham_, Feb 19 2020
%F a(n) = J(n, 9/2), where J(n,x) are the Jacobsthal polynomials. - _G. C. Greubel_, Feb 18 2020
%F E.g.f.: exp(x/2)*(sqrt(37)*cosh(sqrt(37)*x/2) + sinh(sqrt(37)*x/2))/sqrt(37). - _Stefano Spezia_, Feb 19 2020
%p m:=25; S:=series(1/(1-x-9*x^2), x, m+1): seq(coeff(S, x, j), j=0..m); # _G. C. Greubel_, Feb 18 2020
%t CoefficientList[Series[1/(1-x-9*x^2), {x,0,25}], x] (* or *) LinearRecurrence[{1,9}, {1,1}, 25] (* _G. C. Greubel_, Apr 30 2017 *)
%o (Sage) [lucas_number1(n,1,-9) for n in range(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, A128099.
%K nonn,easy
%O 0,3
%A _Olivier Gérard_
%E Edited by _N. J. A. Sloane_, Oct 11 2010
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