A296805 Sum of the larger parts in the partitions of n into two distinct parts with the larger part even.

0, 0, 2, 0, 4, 4, 10, 6, 14, 14, 24, 18, 30, 30, 44, 36, 52, 52, 70, 60, 80, 80, 102, 90, 114, 114, 140, 126, 154, 154, 184, 168, 200, 200, 234, 216, 252, 252, 290, 270, 310, 310, 352, 330, 374, 374, 420, 396, 444, 444, 494, 468, 520, 520, 574, 546, 602, 602

Offset

1,3

Links

Robert Israel, 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 (1,0,0,2,-2,0,0,-1,1).

Formula

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

G.f.: 2*x^3*(1-x+2*x^2+x^4)/((1-x)*(1-x^4)^2). - Robert Israel, Dec 20 2017

From Wesley Ivan Hurt, Jan 15 2024: (Start)

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

a(n) = (1-10*n+6*n^2+(3-10*n)*(-1)^n-2*(2*n+1+(-1)^n)*(-1)^((2*n-1+(-1)^n)/4))/32. (End)

Example

a(12) = 18; the partitions of 12 into two distinct parts are (11,1), (10,2), (9,3), (8,4) and (7,5). The sum of the even numbers among the larger parts gives 10 + 8 = 18.

Maple

f:= gfun:-rectoproc({a(n-9)-a(n-8)-2*a(n-5)+2*a(n-4)+a(n-1)-a(n)=0, seq(a(n)=[0, 0, 2, 0, 4, 4, 10, 6, 14][n], n=1..9)}, a(n), remember):
map(f, [$1..100]); # Robert Israel, Dec 21 2017

Mathematica

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

Crossrefs

Sequence in context: A147980 A021493 A195395 * A336974 A084247 A300307

Adjacent sequences: A296802 A296803 A296804 * A296806 A296807 A296808

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Dec 20 2017

Status

approved