

A295321


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


2



0, 0, 2, 0, 4, 8, 18, 12, 26, 40, 64, 52, 82, 112, 156, 136, 188, 240, 310, 280, 360, 440, 542, 500, 614, 728, 868, 812, 966, 1120, 1304, 1232, 1432, 1632, 1866, 1776, 2028, 2280, 2570, 2460, 2770, 3080, 3432, 3300, 3674, 4048, 4468, 4312, 4756, 5200, 5694
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OFFSET

1,3


COMMENTS

Sum of the areas of the distinct rectangles with even length and integer width such that L + W = n, W < L. For example, a(12) = 52; the rectangles are 2 X 10 and 4 X 8 (6 X 6 is not included since we have W < L), so 2*10 + 4*8 = 52.
Sum of the ordinates from the ordered pairs (nk,n*kk^2) corresponding to integer points along the right side of the parabola b_k = n*kk^2 where nk is an even integer such that 0 < k < floor(n/2).
Sum of the areas of the trapezoids with bases n and n2i and height i for even values of ni where i is in 0 <= i <= floor((n1)/2). For a(n) the area formula for a trapezoid becomes (n+n2i)*i/2 = (2n2i)*i/2 = i*(ni). For n=10, ni is even when i=0,2,4 so a(10) = 0*(100) + 2*(102) + 4*(104) = 0 + 16 + 24 = 40.  Wesley Ivan Hurt, Mar 22 2018
Sum of the areas of the symmetric Lshaped polygons with long side n/2 and width i such that ni is even for i in 0 <= i <= floor((n1)/2). The area of each polygon is given by i^2+2i(n/2i) = i^2+ni2i^2 = i(ni). For n=9, 9i is even for i=1,3 so 1(91) + 3(93) = 8 + 18 = 26.  Wesley Ivan Hurt, Mar 26 2018


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Index entries for sequences related to partitions


FORMULA

a(n) = Sum_{i=1..floor((n1)/2)} i * (ni) * ((ni+1) mod 2).
Conjectures from Colin Barker, Nov 20 2017: (Start)
G.f.: 2*x^3*(1  x + 2*x^2 + 2*x^3 + 2*x^4 + x^6 + x^7) / ((1  x)^4*(1 + x)^3*(1 + x^2)^3).
a(n) = a(n1) + 3*a(n4)  3*a(n5)  3*a(n8) + 3*a(n9) + a(n12)  a(n13) for n>13.
(End)


EXAMPLE

a(16) = 136; the partitions of 16 into two distinct parts are (15,1), (14,2), (13,3), (12,4), (11,5), (10,6), (9,7). There are 3 partitions with the larger part even, and the sum of the products of the smaller and larger parts is then 14*2 + 12*4 + 10*6 = 136.


MAPLE

A295321:=n>add(i*(ni)*((ni+1) mod 2), i=1..floor((n1)/2)): seq(A295321(n), n=1..100);


MATHEMATICA

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


PROG

(PARI) a(n) = sum(i=1, (n1)\2, i*(ni)*((ni+1) % 2)); \\ Michel Marcus, Mar 26 2018


CROSSREFS

Cf. A295320.
Sequence in context: A120710 A115780 A101189 * A001443 A195287 A070015
Adjacent sequences: A295318 A295319 A295320 * A295322 A295323 A295324


KEYWORD

nonn,easy


AUTHOR

Wesley Ivan Hurt, Nov 19 2017


STATUS

approved



