A296161 Sum of the larger parts of the partitions of n into two parts such that the smaller part is odd.

0, 1, 2, 3, 4, 8, 10, 12, 14, 21, 24, 27, 30, 40, 44, 48, 52, 65, 70, 75, 80, 96, 102, 108, 114, 133, 140, 147, 154, 176, 184, 192, 200, 225, 234, 243, 252, 280, 290, 300, 310, 341, 352, 363, 374, 408, 420, 432, 444, 481, 494, 507, 520, 560, 574, 588, 602

Offset

1,3

Comments

Sum of the lengths of the distinct rectangles with integer length and odd width such that L + W = n, W <= L. For example, a(6) = 8; the rectangles with odd width are 1 X 5 and 3 X 3, and the sum of their lengths gives 5 + 3 = 8. - Wesley Ivan Hurt, Dec 06 2017

Links

Table of n, a(n) for n=1..57.

Index entries for sequences related to partitions

Formula

a(n) = Sum_{i=1..floor(n/2)} (n-i) * (i mod 2).

Conjectures from Colin Barker, Dec 06 2017: (Start)

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

a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9) for n>9.

(End)

a(n) = (1/64)*(12*n^2+4*n-6-4*(-1)^((2*n-3-(-1)^n)/4)+(4*n-2)*(-1)^n+(8*n-4)*(-1)^((2*n+3+(-1)^n)/4)). - Wesley Ivan Hurt, Dec 06 2017

Example

a(10) = 21; the partitions of 10 into two parts are 9 + 1, 8 + 2, 7 + 3, 6 + 4 and 5 + 5. Of these, three have an odd smaller part and the sum of the larger parts of these partitions gives 9 + 7 + 5 = 21.

Mathematica

Table[Sum[(n - i)  Mod[i, 2], {i, Floor[n/2]}], {n, 80}]
Table[Total[Select[IntegerPartitions[n, {2}], OddQ[#[[2]]]&][[All, 1]]], {n, 60}] (* Harvey P. Dale, Jan 21 2019 *)

Crossrefs

Cf. A295286, A295287.

Sequence in context: A174816 A161494 A356641 * A121612 A061193 A346971

Adjacent sequences: A296158 A296159 A296160 * A296162 A296163 A296164

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Dec 06 2017

Status

approved