A192426 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.
2, 0, 5, 1, 18, 13, 81, 106, 413, 729, 2258, 4653, 12833, 28666, 74493, 173545, 437346, 1041421, 2583089, 6221322, 15304541, 37079289, 90826994, 220729069, 539487297, 1313161498, 3205831869, 7809748489, 19054635650, 46439068365
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 a(n) = (2, 0, 5, 1, 18, ...) and A192427 = (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); Coefficients(R!( (2-2*x-5*x^2)/(1-x-5*x^2+2*x^3+4*x^4) )); // G. C. Greubel, Jul 12 2023 -
Mathematica
q[x_]:= x+1; d= Sqrt[x^2+8]; p[n_, x_]:= ((x+d)/2)^n + ((x-d)/2)^n (* suggested by A162514 *) Table[Expand[p[n, x]], {n, 0, 6}] reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)}; t= Table[FixedPoint[Expand[#1/. reductionRules] &, p[n,x]], {n,0,30}] Table[Coefficient[Part[t, n], x, 0], {n,30}] (* A192426 *) Table[Coefficient[Part[t, n], x, 1], {n,30}] (* A192427 *) LinearRecurrence[{1,5,-2,-4}, {2,0,5,1}, 40] (* G. C. Greubel, Jul 12 2023 *)
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SageMath
@CachedFunction def a(n): # a = A192426 if (n<4): return (2,0,5,1)[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 12 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.: (2-2*x-5*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-1), where T(n, k) = [x^k] ((x + sqrt(x^2+8))^n + (x - sqrt(x^2+8))^n)/2^n. - G. C. Greubel, Jul 12 2023
Extensions
Typo in name corrected by G. C. Greubel, Jul 12 2023
Comments