A192421 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.
2, 0, 3, 1, 8, 8, 28, 43, 111, 204, 466, 924, 2007, 4109, 8740, 18136, 38240, 79799, 167643, 350664, 735554, 1540104, 3228459, 6762553, 14172272, 29691368, 62217172, 130356451, 273144327, 572305140, 1199164498, 2512579140, 5264623167, 11030890949
Offset: 0
Keywords
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) = 2 + x^2 -> 3 + x. p(3,x) = 3*x + x^3 -> 1 + 5*x. p(4,x) = 2 + 4*x^2 + x^4 -> 8 + 7*x. From these, read a(n) = (2, 0, 3, 1, 8, ...) and A192422 = (0, 1, 1, 5, 7, ...).
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (1,3,-1,-1).
Programs
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Magma
R
:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (2-2*x-3*x^2)/(1-x-3*x^2+x^3+x^4) )); // G. C. Greubel, Jul 11 2023 -
Mathematica
q[x_]:= x+1; d= Sqrt[x^2+4]; p[n_, x_]:= ((x+d)/2)^n + ((x-d)/2)^n (* 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}] (* A192421 *) Table[Coefficient[Part[t, n], x, 1], {n,30}] (* A192422 *) LinearRecurrence[{1,3,-1,-1}, {2,0,3,1}, 40] (* G. C. Greubel, Jul 11 2023 *)
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SageMath
@CachedFunction def a(n): # a = A192421 if (n<4): return (2,0,3,1)[n] else: return a(n-1) +3*a(n-2) -a(n-3) -a(n-4) [a(n) for n in range(41)] # G. C. Greubel, Jul 11 2023
Formula
From Colin Barker, May 12 2014: (Start)
a(n) = a(n-1) + 3*a(n-2) - a(n-3) - a(n-4).
G.f.: (2-2*x-3*x^2)/(1-x-3*x^2+x^3+x^4). (End)
a(n) = Sum_{j=0..n} T(n, j)*Fibonacci(j-1), where T(n, k) = [x^k] ((x + sqrt(x^2+4))^n + (x - sqrt(x^2+4))^n)/2^n. - G. C. Greubel, Jul 11 2023
Comments