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%I #57 Sep 08 2022 08:45:00
%S 1,1,13,25,181,481,2653,8425,40261,141361,624493,2320825,9814741,
%T 37664641,155441533,607417225,2472715621,9761722321,39434309773,
%U 156574977625,629786694901,2508686426401,10066126765213,40170363882025
%N Expansion of 1/((1+3*x)*(1-4*x)).
%C Hankel transform is := 1,12,0,0,0,... - _Philippe Deléham_, Nov 02 2008
%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, 13*a(n-2) equals the number of 13-colored compositions of n with all parts >=2, such that no adjacent parts have the same color. - _Milan Janjic_, Nov 26 2011
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H Vincenzo Librandi, <a href="/A053404/b053404.txt">Table of n, a(n) for n = 0..1000</a>
%H A. Abdurrahman, <a href="https://arxiv.org/abs/1909.10889">CM Method and Expansion of Numbers</a>, arXiv:1909.10889 [math.NT], 2019.
%H F. P. Muga II, <a href="https://www.researchgate.net/publication/267327689_Extending_the_Golden_Ratio_and_the_Binet-de_Moivre_Formula">Extending the Golden Ratio and the Binet-de Moivre Formula</a>, March 2014; Preprint on ResearchGate.
%H A. K. Whitford, <a href="http://www.fq.math.ca/Scanned/15-1/whitford-a.pdf">Binet's formula generalized</a>, Fib. Quart., 15 (1977), pp. 21, 24, 29.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,12).
%F a(n) = ((4^(n+1))-(-3)^(n+1))/7.
%F a(n) = a(n-1) + 12*a(n-2), n > 1; a(0)=1, a(1)=1.
%F From _Paul Barry_, Jul 30 2004: (Start)
%F Convolution of 4^n and (-3)^n.
%F G.f.: 1/((1+3x)(1-4x)); a(n) = Sum_{k=0..n, 4^k*(-3)^(n-k)} = Sum_{k=0..n, (-3)^k*4^(n-k)}. (End)
%F a(n) = Sum_{k, 0<=k<=n} A109466(n,k)*(-12)^(n-k). - _Philippe Deléham_, Oct 26 2008
%F a(n) = (sum_{1<=k<=n+1, k odd} C(n+1,k)*7^(k-1))/2^n. - _Vladimir Shevelev_, Feb 05 2014
%t CoefficientList[Series[1/((1 + 3 x) (1 - 4 x)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 06 2014 *)
%o (Sage) [lucas_number1(n,1,-12) for n in range(1, 25)] # _Zerinvary Lajos_, Apr 22 2009
%o (PARI) a(n)=([0,1; 12,1]^n*[1;1])[1,1] \\ _Charles R Greathouse IV_, Oct 03 2016
%o (Magma) [((4^(n+1)) - (-3)^(n+1))/7: n in [0..30]]; // _G. C. Greubel_, Jan 16 2018
%Y Cf. A001045, A015441.
%K easy,nonn
%O 0,3
%A _Barry E. Williams_, Jan 07 2000
%E More terms from _James A. Sellers_, Feb 02 2000
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