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A226141 Sum of the squared parts of the partitions of n into exactly two parts. 1
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 (list; graph; refs; listen; history; text; internal format)
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/2)^2) * (1 - ceiling(n/2) + floor(n/2)) + Sum_{i=1..n-1} 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

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 *)

CROSSREFS

Sequence in context: A080689 A026321 A288994 * A048221 A183365 A295905

Adjacent sequences:  A226138 A226139 A226140 * A226142 A226143 A226144

KEYWORD

nonn,easy

AUTHOR

Wesley Ivan Hurt, May 27 2013

STATUS

approved

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Last modified September 17 19:06 EDT 2021. Contains 347489 sequences. (Running on oeis4.)