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A053404 Expansion of 1/((1+3*x)*(1-4*x)).

%I A053404 #66 Jul 12 2025 18:40:19
%S A053404 1,1,13,25,181,481,2653,8425,40261,141361,624493,2320825,9814741,
%T A053404 37664641,155441533,607417225,2472715621,9761722321,39434309773,
%U A053404 156574977625,629786694901,2508686426401,10066126765213,40170363882025
%N A053404 Expansion of 1/((1+3*x)*(1-4*x)).
%C A053404 Hankel transform is := 1,12,0,0,0,... - _Philippe Deléham_, Nov 02 2008
%C A053404 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 A053404 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H A053404 Vincenzo Librandi, <a href="/A053404/b053404.txt">Table of n, a(n) for n = 0..1000</a>
%H A053404 A. Abdurrahman, <a href="https://arxiv.org/abs/1909.10889">CM Method and Expansion of Numbers</a>, arXiv:1909.10889 [math.NT], 2019.
%H A053404 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 A053404 A. K. Whitford, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/15-1/whitford-a.pdf">Binet's formula generalized</a>, Fib. Quart., 15 (1977), pp. 21, 24, 29.
%H A053404 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,12).
%F A053404 a(n) = ((4^(n+1))-(-3)^(n+1))/7.
%F A053404 a(n) = a(n-1) + 12*a(n-2), n > 1; a(0)=1, a(1)=1.
%F A053404 From _Paul Barry_, Jul 30 2004: (Start)
%F A053404 Convolution of 4^n and (-3)^n.
%F A053404 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 A053404 a(n) = Sum_{k, 0<=k<=n} A109466(n,k)*(-12)^(n-k). - _Philippe Deléham_, Oct 26 2008
%F A053404 a(n) = (sum_{1<=k<=n+1, k odd} C(n+1,k)*7^(k-1))/2^n. - _Vladimir Shevelev_, Feb 05 2014
%F A053404 From _Peter Bala_, Jun 27 2025: (Start)
%F A053404 a(n) = hypergeom([1/2 - (1/2)*n, -(1/2)*n], [-n], -48) for n >= 1.
%F A053404 The following products telescope:
%F A053404 Product_{k >= 0} (1 + 12^k/a(2*k+1)) = 8.
%F A053404 Product_{k >= 1} (1 - 12^k/a(2*k+1)) = 4/25.
%F A053404 Product_{k >= 0} (1 + (-12)^k/a(2*k+1)) = 8/7.
%F A053404 Product_{k >= 1} (1 - (-12)^k/a(2*k+1)) = 28/25. (End)
%p A053404 seq(simplify(hypergeom([1/2 - (1/2)*n, -(1/2)*n], [-n], -48)), n = 1..40); # _Peter Bala_, Jul 05 2025
%t A053404 CoefficientList[Series[1/((1 + 3 x) (1 - 4 x)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 06 2014 *)
%o A053404 (Sage) [lucas_number1(n,1,-12) for n in range(1, 25)] # _Zerinvary Lajos_, Apr 22 2009
%o A053404 (PARI) a(n)=([0,1; 12,1]^n*[1;1])[1,1] \\ _Charles R Greathouse IV_, Oct 03 2016
%o A053404 (Magma) [((4^(n+1)) - (-3)^(n+1))/7: n in [0..30]]; // _G. C. Greubel_, Jan 16 2018
%Y A053404 Cf. A000045, A001045, A006130, A006131, A015440 - A015443, A015445 - A015447, A053404, A053428, A168579, A350468, A350469.
%K A053404 easy,nonn
%O A053404 0,3
%A A053404 _Barry E. Williams_, Jan 07 2000
%E A053404 More terms from _James Sellers_, Feb 02 2000