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A024862
a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = natural numbers, t = odd natural numbers.
2
3, 5, 17, 23, 50, 62, 110, 130, 205, 235, 343, 385, 532, 588, 780, 852, 1095, 1185, 1485, 1595, 1958, 2090, 2522, 2678, 3185, 3367, 3955, 4165, 4840, 5080, 5848, 6120, 6987, 7293, 8265, 8607, 9690, 10070, 11270, 11690, 13013, 13475, 14927, 15433, 17020, 17572, 19300
OFFSET
2,1
FORMULA
G.f.: x^2*(3+2*x+3*x^2) / ((1+x)^3*(x-1)^4). - R. J. Mathar, Sep 25 2013
a(n) = 3*A058187(n-2) + 2*A058187(n-3) + 3*A058187(n-4). - R. J. Mathar, Sep 25 2013
From Colin Barker, Jan 29 2016: (Start)
a(n) = (8*n^3 + 6*(-1)^n*n^2 + 12*n^2 + 6*(-1)^n*n - 2*n + 3*(-1)^n - 3)/48.
a(n) = (4*n^3 + 9*n^2 + 2*n)/24 for n even.
a(n) = (4*n^3 + 3*n^2 - 4*n - 3)/24 for n odd. (End)
E.g.f.: (1/48)*(3*(1 - 4*x + 2*x^2)*exp(-x) + (-3 + 18*x + 36*x^2 + 8*x^3)*exp(x)). - G. C. Greubel, Apr 19 2023
MATHEMATICA
CoefficientList[Series[(3+2x+3x^2)/((1+x)^3 (1-x)^4), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 25 2013 *)
PROG
(PARI) Vec(x^2*(3+2*x+3*x^2)/((1+x)^3*(x-1)^4) + O(x^100)) \\ Colin Barker, Jan 29 2016
(Magma) [((2*n-1)*(2*n+1)*(2*n+3) +3*(-1)^n*(n^2+(n+1)^2))/48: n in [2..50]]; // G. C. Greubel, Apr 19 2023
(SageMath) [((2*n-1)*(2*n+1)*(2*n+3) +3*(-1)^n*(n^2+(n+1)^2))/48 for n in range(2, 51)] # G. C. Greubel, Apr 19 2023
CROSSREFS
Cf. A058187.
Sequence in context: A100564 A231232 A154608 * A025106 A333199 A203193
KEYWORD
nonn,easy
STATUS
approved