A024312 a(n) = s(1)*s(n) + s(2)*s(n-1) + ... + s(k)*s(n+1-k), where k = floor((n+1)/2), s = (natural numbers >= 3).
9, 12, 31, 38, 70, 82, 130, 148, 215, 240, 329, 362, 476, 518, 660, 712, 885, 948, 1155, 1230, 1474, 1562, 1846, 1948, 2275, 2392, 2765, 2898, 3320, 3470, 3944, 4112, 4641, 4828, 5415, 5622, 6270, 6498, 7210, 7460, 8239, 8512, 9361, 9658, 10580, 10902, 11900, 12248, 13325
Offset: 1
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).
Crossrefs
Programs
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Magma
[(&+[j*(n+5-j): j in [3..Floor((n+5)/2)]]) : n in [1..50]]; // G. C. Greubel, Jan 17 2022
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Mathematica
Table[Sum[j*(n-j+5), {j, 3, Floor[(n+5)/2]}], {n, 50}] (* G. C. Greubel, Jan 17 2022 *) LinearRecurrence[{1,3,-3,-3,3,1,-1},{9,12,31,38,70,82,130},60] (* Harvey P. Dale, Aug 23 2024 *) -
PARI
a(n)=((75+182*n+63*n^2+4*n^3)-3*(25+10*n+n^2)*(-1)^n)/48 \\ Charles R Greathouse IV, Oct 21 2022
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Sage
[( (75 +182*n +63*n^2 +4*n^3) - 3*(25 +10*n +n^2)*(-1)^n )/48 for n in (1..50)] # G. C. Greubel, Jan 17 2022
Formula
From G. C. Greubel, Jan 17 2022: (Start)
a(n) = ( (75 + 182*n + 63*n^2 + 4*n^3) - 3*(25 + 10*n + n^2)*(-1)^n )/48.
G.f.: x*(9 + 3*x - 8*x^2 - 2*x^3 + 2*x^4)/((1-x)^4 * (1+x)^3).
a(n) = (-60 - 18*n + (14 + 3*n)*f(n) + 3*(4+n)*f(n)^2 - 2*f(n)^3)/6, where f(n) = floor((n+5)/2). (End)