A192427 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.
0, 1, 1, 8, 11, 45, 80, 251, 517, 1432, 3195, 8317, 19360, 48827, 116213, 288360, 694331, 1708397, 4138480, 10138363, 24636645, 60217912, 146570491, 357833309, 871703360, 2126857275, 5183425493, 12642971912, 30819571387, 75160150861
Offset: 0
Examples
The first five polynomials p(n,x) and their reductions are as follows: p(0,x) = 2 -> 2 p(1,x) = x -> x p(2,x) = 4 + x^2 -> 5 + x p(3,x) = 6*x + x^3 -> 1 + 8*x p(4,x) = 8 + 8*x^2 + x^4 -> 18 + 11*x. From these, read A192426 = (2, 0, 5, 1, 18, ...) and a(n) = (0, 1, 1, 8, 11, ...).
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (1,5,-2,-4).
Programs
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Magma
R
:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1+2*x^2)/(1-x-5*x^2+2*x^3+4*x^4) )); // G. C. Greubel, Jul 13 2023 -
Mathematica
(See A192426.) LinearRecurrence[{1,5,-2,-4}, {0,1,1,8}, 40] (* G. C. Greubel, Jul 13 2023 *)
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SageMath
@CachedFunction def a(n): # a = A192427 if (n<4): return (0,1,1,8)[n] else: return a(n-1) + 5*a(n-2) - 2*a(n-3) - 4*a(n-4) [a(n) for n in range(41)] # G. C. Greubel, Jul 13 2023
Formula
From Colin Barker, May 12 2014: (Start)
a(n) = a(n-1) + 5*a(n-2) - 2*a(n-3) - 4*a(n-4).
G.f.: x*(1+2*x^2)/(1-x-5*x^2+2*x^3+4*x^4). (End)
a(n) = Sum_{k=0..n} T(n, k)*Fibonacci(k), where T(n, k) = [x^k] ((x + sqrt(x^2+8))^n + (x - sqrt(x^2+8))^n)/2^n. - G. C. Greubel, Jul 13 2023
Extensions
Typo in name corrected by G. C. Greubel, Jul 13 2023
Comments