A226141 Sum of the squared parts of the partitions of n into exactly two parts.

0, 2, 5, 18, 30, 64, 91, 156, 204, 310, 385, 542, 650, 868, 1015, 1304, 1496, 1866, 2109, 2570, 2870, 3432, 3795, 4468, 4900, 5694, 6201, 7126, 7714, 8780, 9455, 10672, 11440, 12818, 13685, 15234, 16206, 17936, 19019, 20940, 22140, 24262, 25585, 27918, 29370, 31924, 33511

Offset: 1

Author

Wesley Ivan Hurt, May 27 2013

Keywords

• nonn
• easy

Examples

			a(5) = 30; 5 has exactly 2 partitions into two parts, (4,1) and (3,2). Squaring the parts and adding, we get: 1^2 + 2^2 + 3^2 + 4^2 = 30.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to partitions
• Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-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), this sequence (k=2), A294270 (k=3), A294271 (k=4), A294272 (k=5), A294273 (k=6), A294274 (k=7), A294275 (k=8), A294276 (k=9), A294279 (k=10).

Cf. A000330, A308422.

Programs

Magma

[n*(8*n^2 - 9*n + 4)/24 + (-1)^n*n^2/8 : n in [1..80]]; // Wesley Ivan Hurt, Jun 22 2024

Maple

a:=n->sum(i^2 + (n-i)^2, i=1..floor(n/2)); seq((a(k), k=1..40);

Mathematica

Array[Sum[i^2 + (# - i)^2, {i, Floor[#/2]}] &, 39] (* Michael De Vlieger, Jan 23 2018 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{0,2,5,18,30,64,91},50] (* Harvey P. Dale, Jul 23 2019 *)

Formula

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

a(n) = n*(8*n^2 - 9*n + 4)/24 + (-1)^n*n^2/8. - Giovanni Resta, May 29 2013

G.f.: x^2*(2+3*x+7*x^2+3*x^3+x^4) / ( (1+x)^3*(x-1)^4 ). - R. J. Mathar, Jun 07 2013

a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7). - Wesley Ivan Hurt, Jun 22 2024

a(n) = A000330(n) - A308422(n). - Wesley Ivan Hurt, Jul 16 2025