A104237 Expansion of (1 - x + x^2)*(1 + x + x^2 - x^3 + 2*x^4)/((1 - x)*(1 + x)^2*(1 + x^2)*(1 + x - x^2 + x^3)).
1, -2, 5, -11, 26, -53, 104, -198, 375, -700, 1299, -2401, 4432, -8167, 15038, -27676, 50925, -93686, 172337, -316999, 583078, -1072473, 1972612, -3628226, 6673379, -12274288, 22575967, -41523709, 76374044, -140473803, 258371642
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (-2,0,0,0,2,0,0,1).
Programs
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Mathematica
LinearRecurrence[{-2,0,0,0,2,0,0,1},{1,-2,5,-11,26,-53,104,-198},40] (* Harvey P. Dale, May 07 2016 *) -
PARI
Vec((1 - x + x^2)*(1 + x + x^2 - x^3 + 2*x^4) / ((1 - x)*(1 + x)^2*(1 + x^2)*(1 + x - x^2 + x^3)) + O(x^40)) \\ Colin Barker, May 21 2019
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SageMath
@CachedFunction def A000073(n): if (n<3): return (n//2) else: return A000073(n-1) + A000073(n-2) + A000073(n-3) def A104237(n): return (1/4)*(2*(-1)^n*(A000073(n+5) + A000073(n+4)) - 2*i^(n-1)*(n%2) - 3*(-1)^n*(2*n+3) + 1) [A104237(n) for n in (0..60)] # G. C. Greubel, Jul 08 2022
Formula
G.f.: (1 - x + x^2)*(1 + x + x^2 - x^3 + 2*x^4) / ((1 - x)*(1 + x)^2*(1 + x^2)*(1 + x - x^2 + x^3)).
a(n) = -2*a(n-1) + 2*a(n-5) + a(n-8) for n>7. - Colin Barker, May 21 2019
a(n) = (1/4)*(2*(-1)^n*(A000073(n+5) + A000073(n+4)) - 2*A056594(n-1) - 3*(-1)^n*(2*n+3) - 1). - G. C. Greubel, Jul 08 2022
Comments