A294270 - OEIS

%I #35 Feb 03 2026 03:27:20

%S 0,2,9,44,100,252,441,848,1296,2150,3025,4572,6084,8624,11025,14912,

%T 18496,24138,29241,37100,44100,54692,64009,77904,90000,107822,123201,

%U 145628,164836,192600,216225,250112,278784,319634,354025,402732,443556,501068,549081

%N Sum of the cubes of the parts in the partitions of n into two parts.

%C 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

%H Colin Barker, <a href="/A294270/b294270.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-4,-6,6,4,-4,-1,1).

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

%F From _Colin Barker_, Nov 20 2017: (Start)

%F 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).

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

%F 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)

%F 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

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

%o (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

%o (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

%Y 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).

%K nonn,easy

%O 1,2

%A _Wesley Ivan Hurt_, Oct 26 2017