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

1,2

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).

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

Example

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.

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 *)

Prog

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

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.

Sequence in context: A080689 A026321 A288994 * A355515 A048221 A183365

Adjacent sequences: A226138 A226139 A226140 * A226142 A226143 A226144

Keyword

nonn,easy

Author

Wesley Ivan Hurt, May 27 2013

Status

approved