A274324 Number of partitions of n^3 into at most two parts.
1, 1, 5, 14, 33, 63, 109, 172, 257, 365, 501, 666, 865, 1099, 1373, 1688, 2049, 2457, 2917, 3430, 4001, 4631, 5325, 6084, 6913, 7813, 8789, 9842, 10977, 12195, 13501, 14896, 16385, 17969, 19653, 21438, 23329, 25327, 27437, 29660, 32001, 34461, 37045, 39754
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).
Programs
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Magma
[(3+(-1)^n+2*n^3)/4 : n in [0..50]]; // Wesley Ivan Hurt, Jun 25 2016
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Maple
A274324:=n->(3+(-1)^n+2*n^3)/4: seq(A274324(n), n=0..50); # Wesley Ivan Hurt, Jun 25 2016
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Mathematica
Table[(3+(-1)^n+2*n^3)/4, {n, 0, 50}] (* Wesley Ivan Hurt, Jun 25 2016 *) -
PARI
\\ b(n) is the coefficient of x^n in the g.f. 1/((1-x)*(1-x^2)). b(n) = (3+(-1)^n+2*n)/4 vector(50, n, n--; b(n^3))
Formula
Coefficient of x^(n^3) in 1/((1-x)*(1-x^2)).
a(n) = A008619(n^3).
a(n) = (3+(-1)^n+2*n^3)/4.
a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5) for n>4.
G.f.: (1-2*x+4*x^2+3*x^3) / ((1-x)^4*(1+x)).
From Stefano Spezia, Sep 28 2022: (Start)
a(n) = A050492((n+1)/2) for n odd.
E.g.f.: ((2 + x + 3*x^2 + x^3)*cosh(x) + (1 + x + 3*x^2 + x^3)*sinh(x))/2. (End)
Comments