A294270 Sum of the cubes of the parts in the partitions of n into two parts.

0, 2, 9, 44, 100, 252, 441, 848, 1296, 2150, 3025, 4572, 6084, 8624, 11025, 14912, 18496, 24138, 29241, 37100, 44100, 54692, 64009, 77904, 90000, 107822, 123201, 145628, 164836, 192600, 216225, 250112, 278784, 319634, 354025, 402732, 443556, 501068, 549081

Offset

1,2

Comments

a(n) is a square when n is odd. In fact: a(2*k+1) = (2*k^2 + k)^2; a(2*k) = k^2*(4*k^2 - 3*k + 1), where (2*k)^2 > 4*k^2 - 3*k + 1 > (2*k - 1)^2 for k>0. - Bruno Berselli, Nov 20 2017

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Formula

a(n) = Sum_{i=1..floor(n/2)} i^3 + (n-i)^3.

From Colin Barker, Nov 20 2017: (Start)

G.f.: x^2*(2 + 7*x + 27*x^2 + 28*x^3 + 24*x^4 + 7*x^5 + x^6) / ((1 - x)^5*(1 + x)^4).

a(n) = n^2*(4*n^2 - 7*n + 4 + n*(-1)^n)/16.

a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n>9. (End)

E.g.f.: x*(x*(7 + 8*x + 2*x^2)*cosh(x) + (1 + 4*x + 9*x^2 + 2*x^3)*sinh(x))/8. - Stefano Spezia, Feb 03 2026

Mathematica

Table[Sum[i^3 + (n - i)^3, {i, Floor[n/2]}], {n, 80}]

Prog

(PARI) concat(0, Vec(x^2*(2 + 7*x + 27*x^2 + 28*x^3 + 24*x^4 + 7*x^5 + x^6) / ((1 - x)^5*(1 + x)^4) + O(x^40))) \\ Colin Barker, Nov 20 2017
(Magma) [0] cat &cat[[k^2*(4*k^2-3*k+1), k^2*(2*k+1)^2]: k in [1..20]]; // Bruno Berselli, Nov 22 2017

Crossrefs

Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), A226141 (k=2), this sequence (k=3), A294271 (k=4), A294272 (k=5), A294273 (k=6), A294274 (k=7), A294275 (k=8), A294276 (k=9), A294279 (k=10).

Sequence in context: A345414 A272199 A260074 * A163650 A259777 A365038

Adjacent sequences: A294267 A294268 A294269 * A294271 A294272 A294273

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 26 2017

Status

approved