A294286 Sum of the squares of the parts in the partitions of n into two distinct parts.

0, 0, 5, 10, 30, 46, 91, 124, 204, 260, 385, 470, 650, 770, 1015, 1176, 1496, 1704, 2109, 2370, 2870, 3190, 3795, 4180, 4900, 5356, 6201, 6734, 7714, 8330, 9455, 10160, 11440, 12240, 13685, 14586, 16206, 17214, 19019, 20140, 22140, 23380, 25585, 26950, 29370

Offset

1,3

Links

Iain Fox, Table of n, a(n) for n = 1..10000 (first 1000 terms from Colin Barker)

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-1)/2)} i^2 + (n-i)^2.

From David A. Corneth, Oct 27 2017: (Start)

For odd n, a(n) = n^3/3 - n^2/2 + n/6 = A000330(n + 1).

For even n, a(n) = n^3/3 - 3*n^2/4 + n/6.

(End)

From Colin Barker, Nov 04 2017: (Start)

G.f.: x^3*(5 + 5*x + 5*x^2 + x^3) / ((1 - x)^4*(1 + x)^3).

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) for n > 7.

(End)

a(n) = n*(2*n^2-3*n*(5+(-1)^n)/4+1)/6. - Wesley Ivan Hurt, Dec 03 2023

Example

For n = 6, there are two ways of partitioning 6 into two distinct parts: 6 = 1+5 and 6 = 2+4.  So a(6) = 1^2 + 5^2 + 2^2 + 4^2 = 46.
For n = 7, there are three ways of partitioning 7 into two distinct parts: 7 = 1+6, 7 = 2+5, and 7 = 3+4.  So a(7) = 1^2 + 6^2 + 2^2 + 5^2 + 3^2 + 4^2 = 91. - Michael B. Porter, Nov 05 2017

Mathematica

Table[Sum[i^2 + (n - i)^2, {i, Floor[(n-1)/2]}], {n, 40}]
Table[Total[Flatten[Select[IntegerPartitions[n, {2}], #[[1]]!=#[[2]]&]]^2], {n, 50}] (* Harvey P. Dale, Dec 02 2022 *)

Prog

(PARI) first(n) = my(res = vector(n, i, i^3 / 3 - i^2 / 2 + i / 6)); forstep(i = 2, n, 2, res[i] -= i^2 >> 2); res \\ David A. Corneth, Oct 27 2017
(PARI) concat(vector(2), Vec(x^3*(5 + 5*x + 5*x^2 + x^3) / ((1 - x)^4*(1 + x)^3) + O(x^60))) \\ Colin Barker, Nov 04 2017
(Magma) [n*(2*n^2-3*n*(5+(-1)^n)/4+1)/6 : n in [1..60]]; // Wesley Ivan Hurt, Dec 03 2023

Crossrefs

Cf. A000330.

Sequence in context: A005514 A069921 A053818 * A133629 A156302 A156234

Adjacent sequences: A294283 A294284 A294285 * A294287 A294288 A294289

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 26 2017

Status

approved