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

Author

Wesley Ivan Hurt, Oct 26 2017

Keywords

• nonn
• easy

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).

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).

Programs

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

Mathematica

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

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

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)