A192384 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.
0, 2, 4, 24, 72, 312, 1088, 4288, 15744, 60192, 224832, 851072, 3197056, 12062592, 45398016, 171104256, 644354048, 2427699712, 9144222720, 34448209920, 129761986560, 488821962752, 1841370087424, 6936475090944, 26129575084032
Offset: 1
Examples
The first five polynomials p(n,x) and their reductions are as follows: p(0, x) = 1 -> 1 p(1, x) = 2*x -> 2*x p(2, x) = 3 + x + 3*x^2 -> 6 + 4*x p(3, x) = 12*x + 4*x^2 + 4*x^3 -> 8 + 24*x p(4, x) = 9 + 6*x + 31*x^2 + 10*x^3 + 5*x^4 -> 60 + 72*x. From these, read A192383 = (1, 0, 6, 8, 60, ...) and A192384 = (0, 2, 4, 24, 72, ...).
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (2,8,-4,-4).
Programs
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Magma
R
:=PowerSeriesRing(Integers(), 41); [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 -
Mathematica
(See A192383.) LinearRecurrence[{2,8,-4,-4}, {0,2,4,24}, 40] (* G. C. Greubel, Jul 10 2023 *)
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SageMath
@CachedFunction def a(n): # a = A192384 if (n<5): return (0,0,2,4,24)[n] else: return 2*a(n-1) +8*a(n-2) -4*a(n-3) -4*a(n-4) [a(n) for n in range(1,41)] # G. C. Greubel, Jul 10 2023
Formula
From Colin Barker, Dec 09 2012: (Start)
a(n) = 2*a(n-1) + 8*a(n-2) - 4*a(n-3) - 4*a(n-4).
G.f.: 2*x^2/(1-2*x-8*x^2+4*x^3+4*x^4). (End)
From G. C. Greubel, Jul 10 2023: (Start)
T(n, k) = [x^k] ((x+sqrt(x+3))^n - (x-sqrt(x+3))^n)/(2*sqrt(x+3)).
a(n) = Sum_{k=0..n-1} T(n, k)*Fibonacci(k). (End)
Comments