A272104 Sum of the even numbers among the larger parts of the partitions of n into two parts.

0, 0, 0, 2, 2, 4, 4, 10, 10, 14, 14, 24, 24, 30, 30, 44, 44, 52, 52, 70, 70, 80, 80, 102, 102, 114, 114, 140, 140, 154, 154, 184, 184, 200, 200, 234, 234, 252, 252, 290, 290, 310, 310, 352, 352, 374, 374, 420, 420, 444, 444, 494, 494, 520, 520, 574, 574, 602

Offset: 0

Author

Wesley Ivan Hurt, Apr 20 2016

Keywords

• nonn
• easy

Comments

Essentially, repeated values of A152749.

Sum of the lengths of the distinct rectangles with even length and integer width such that L + W = n, W <= L. For example, a(10) = 14; the rectangles are 2 X 8 and 4 X 6, so 8 + 6 = 14. - Wesley Ivan Hurt, Nov 04 2017

Examples

			a(5) = 4; the partitions of 5 into 2 parts are (4,1),(3,2) and the sum of the larger even parts is 4.
a(6) = 4; the partitions of 6 into 2 parts are (5,1),(4,2),(3,3) and the sum of the larger even parts is also 4.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• 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).

Crossrefs

Cf. A000326, A001318, A026741, A049450, A049452, A109043, A126964, A152749, A268351, A272212.

Programs

Magma

[(1+3*(2*n-3-(-1)^n)/2+3*(2*n-3-(-1)^n)^2/8+(2*n-1-(-1)^n)*(-1)^((2*n+1-(-1)^n) div 4)/2)/8 : n in [0..50]];

Maple

A272104:=n->(1+3*(2*n-3-(-1)^n)/2+3*(2*n-3-(-1)^n)^2/8+(2*n-1-(-1)^n)*(-1)^((2*n+1-(-1)^n)/4)/2)/8: seq(A272104(n), n=0..100);

Mathematica

Table[(1 + 3(2n-3-(-1)^n)/2 + 3(2n-3-(-1)^n)^2/8 + (2n-1-(-1)^n) * (-1)^((2n+1-(-1)^n)/4)/2) / 8, {n, 0, 50}]
Table[Total@ Map[First, IntegerPartitions[n, {2}] /. {k_, } /; OddQ@ k -> Nothing], {n, 0, 57}] (* _Michael De Vlieger, Apr 20 2016, Version 10.2 *)

PARI

concat(vector(3), Vec(2*x^3*(1-x+x^2)*(1+x+x^2)/((1-x)^3*(1+x)^2*(1+x^2)^2) + O(x^50))) \\ Colin Barker, Apr 20 2016

Formula

a(n) = (1 + 3*(2n-3-(-1)^n)/2 + 3*(2n-3-(-1)^n)^2/8 + (2n-1-(-1)^n) * (-1)^((2n+1-(-1)^n)/4)/2) / 8.

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

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

a(2n+1) = a(2n+2) = A152749(n) = 2*A001318(n).

G.f.: 2*x^3*(1-x+x^2)*(1+x+x^2) / ((1-x)^3*(1+x)^2*(1+x^2)^2). - Colin Barker, Apr 20 2016

From Wesley Ivan Hurt, Apr 22 2016, Apr 23 2016: (Start)

a(2n+2)-a(2n) = A109043(n) = 2*A026741(n).

a(4n) = A049450(n) = 2*A000326(n),

a(8n) = A126964(n) = 2*A049452(n),

a(12n) = 2*A268351(n).

a(n+1) = A001318(n) - A272212(n+1). (End)

E.g.f.: ((2 + 3*x*(1 + x))*cosh(x) - 2*(cos(x) + x*cos(x) + x*sin(x)) + (-1 + 3*(-1 + x)*x)*sinh(x))/16. - Ilya Gutkovskiy, Apr 29 2016