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.

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

Author

Clark Kimberling

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 2..1000
• Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Crossrefs

Cf. A058187.

Programs

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

Mathematica

CoefficientList[Series[(3+2x+3x^2)/((1+x)^3 (1-x)^4), {x,0,50}], x] (* Vincenzo Librandi, Sep 25 2013 *)

PARI

Vec(x^2*(3+2*x+3*x^2)/((1+x)^3*(x-1)^4) + O(x^100)) \\ Colin Barker, Jan 29 2016

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

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