A024868 a(n) = 2*(n+1) + 3*n + ... + (k+1)*(n+2-k), where k = floor(n/2).
6, 8, 22, 27, 52, 61, 100, 114, 170, 190, 266, 293, 392, 427, 552, 596, 750, 804, 990, 1055, 1276, 1353, 1612, 1702, 2002, 2106, 2450, 2569, 2960, 3095, 3536, 3688, 4182, 4352, 4902, 5091, 5700, 5909, 6580, 6810, 7546, 7798, 8602, 8877, 9752, 10051, 11000, 11324, 12350
Offset: 2
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).
Programs
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Magma
[19*n/24-9/16+n^3/12+11*n^2/16+(-1)^n*(3*n/8 +9/16+n^2/16): n in [2..50]]; // Vincenzo Librandi, Sep 26 2013
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Maple
seq(sum((i+1)*(k-i+2), i=1..floor(k/2)), k=2..70); # Wesley Ivan Hurt, Sep 20 2013
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Mathematica
Table[Floor[n/2] (-2Floor[n/2]^2 +3n*Floor[n/2] +9n +14)/6, {n, 2, 100}] (* Wesley Ivan Hurt, Sep 20 2013 *) CoefficientList[Series[(6 +2x -4x^2 -x^3 +x^4)/((1+x)^3 (1-x)^4), {x, 0, 60}], x] (* Vincenzo Librandi, Sep 26 2013 *) LinearRecurrence[{1,3,-3,-3,3,1,-1},{6,8,22,27,52,61,100},50] (* Harvey P. Dale, Aug 11 2023 *) -
SageMath
[(1/48)*(4*n^3 +33*n^2 +38*n -27 +3*(-1)^n*(n+3)^2) for n in (2..60)] # G. C. Greubel, Jul 13 2022
Formula
a(n) = Sum_{i=1..floor(n/2)} (i+1)*(n-i+2) = floor(n/2)*(-2*floor(n/2)^2 + 3*n*floor(n/2) + 9*n + 14)/6, n>1. - Wesley Ivan Hurt, Sep 20 2013
G.f.: x^2*(6 + 2*x - 4*x^2 - x^3 + x^4) / ( (1+x)^3*(x-1)^4 ). - R. J. Mathar, Sep 25 2013
a(n) = 6*A058187(n-2) +2*A058187(n-3) -4*A058187(n-4) -A058187(n-5) +A058187(n-6). - R. J. Mathar, Sep 25 2013
a(n) = ( 4*n^3 + 33*n^2 + 38*n - 27 )/48 + (-1)^n*(n+3)^2/16. - R. J. Mathar, Sep 25 2013
E.g.f.: (1/24)*( x*(2*x^2 + 24*x + 27)*cosh(x) + (2*x^3 + 21*x^2 + 48*x - 27)*sinh(x) ). - G. C. Greubel, Jul 13 2022