|
%I #16 Jul 10 2023 08:21:14
%S 0,2,4,24,72,312,1088,4288,15744,60192,224832,851072,3197056,12062592,
%T 45398016,171104256,644354048,2427699712,9144222720,34448209920,
%U 129761986560,488821962752,1841370087424,6936475090944,26129575084032
%N Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.
%C Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.
%C The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2*d), where d = sqrt(x+3). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232.
%H G. C. Greubel, <a href="/A192384/b192384.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,8,-4,-4).
%F From _Colin Barker_, Dec 09 2012: (Start)
%F a(n) = 2*a(n-1) + 8*a(n-2) - 4*a(n-3) - 4*a(n-4).
%F G.f.: 2*x^2/(1-2*x-8*x^2+4*x^3+4*x^4). (End)
%F From _G. C. Greubel_, Jul 10 2023: (Start)
%F T(n, k) = [x^k] ((x+sqrt(x+3))^n - (x-sqrt(x+3))^n)/(2*sqrt(x+3)).
%F a(n) = Sum_{k=0..n-1} T(n, k)*Fibonacci(k). (End)
%e The first five polynomials p(n,x) and their reductions are as follows:
%e p(0, x) = 1 -> 1
%e p(1, x) = 2*x -> 2*x
%e p(2, x) = 3 + x + 3*x^2 -> 6 + 4*x
%e p(3, x) = 12*x + 4*x^2 + 4*x^3 -> 8 + 24*x
%e p(4, x) = 9 + 6*x + 31*x^2 + 10*x^3 + 5*x^4 -> 60 + 72*x.
%e From these, read A192383 = (1, 0, 6, 8, 60, ...) and A192384 = (0, 2, 4, 24, 72, ...).
%t (See A192383.)
%t LinearRecurrence[{2,8,-4,-4}, {0,2,4,24}, 40] (* _G. C. Greubel_, Jul 10 2023 *)
%o (Magma)
%o R<x>:=PowerSeriesRing(Integers(), 41);
%o [0] cat Coefficients(R!( 2*x^2/(1-2*x-8*x^2+4*x^3+4*x^4) )) // _G. C. Greubel_, Jul 10 2023
%o (SageMath)
%o @CachedFunction
%o def a(n): # a = A192384
%o if (n<5): return (0,0,2,4,24)[n]
%o else: return 2*a(n-1) +8*a(n-2) -4*a(n-3) -4*a(n-4)
%o [a(n) for n in range(1,41)] # _G. C. Greubel_, Jul 10 2023
%Y Cf. A192232, A192383, A192385.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Jun 30 2011
|
