A296141 Sum of the smaller parts of the partitions of n into two distinct parts with the larger part even.

0, 0, 1, 0, 1, 2, 4, 2, 4, 6, 9, 6, 9, 12, 16, 12, 16, 20, 25, 20, 25, 30, 36, 30, 36, 42, 49, 42, 49, 56, 64, 56, 64, 72, 81, 72, 81, 90, 100, 90, 100, 110, 121, 110, 121, 132, 144, 132, 144, 156, 169, 156, 169, 182, 196, 182, 196, 210, 225, 210, 225, 240

Offset

1,6

Comments

a(n+1) is the sum of the smaller parts in the partitions of n into two parts with the larger part odd. For example, a(11) = 9; the partitions of 10 into two parts are (9,1), (8,2), (7,3), (6,4) and (5,5). Three of these partitions have an odd number as their larger part, namely (9,1), (7,3) and (5,5). Adding the smaller parts of these partitions gives 1 + 3 + 5 = 9.

Links

Table of n, a(n) for n=1..62.

Index entries for sequences related to partitions

Formula

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

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

G.f.: x^3*(1 - x + x^2 + x^3) / ((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) = floor((n+1)/4)^2*(n mod 2)+(1+floor((n-2)/4))*floor((n-2)/4)*((n+1) mod 2). - Wesley Ivan Hurt, Dec 08 2017

Example

a(10) = 6; the partitions of 10 into two parts are (9,1), (8,2), (7,3), (6,4) and (5,5). Two of these partitions have an even number as their larger part, namely (8,2) and (6,4). Adding the smaller parts of these partitions gives 2 + 4 = 6.

Mathematica

Table[Sum[i Mod[n - i + 1, 2], {i, Floor[(n - 1)/2]}], {n, 80}]

Prog

(PARI) a(n) = sum(i=1, floor((n-1)/2), i*lift(Mod(n-i+1, 2))) \\ Iain Fox, Dec 06 2017

Crossrefs

Cf. A295287, A295293.

Sequence in context: A047975 A112791 A227346 * A286536 A318768 A166242

Adjacent sequences: A296138 A296139 A296140 * A296142 A296143 A296144

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Dec 05 2017

Status

approved