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A295317 Sum of the products of the smaller and larger parts of the partitions of n into two distinct parts with the smaller part odd. 2
0, 0, 2, 3, 4, 5, 18, 22, 26, 30, 64, 73, 82, 91, 156, 172, 188, 204, 310, 335, 360, 385, 542, 578, 614, 650, 868, 917, 966, 1015, 1304, 1368, 1432, 1496, 1866, 1947, 2028, 2109, 2570, 2670, 2770, 2870, 3432, 3553, 3674, 3795, 4468, 4612, 4756, 4900, 5694 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Sum of the areas of the distinct rectangles with integer length and odd width such that L + W = n, W < L. For example, a(10) = 30; the rectangles are 1 X 9 and 3 X 7 (5 X 5 is not included since we have W < L), so 1*9 + 3*7 = 30.

Sum of the ordinates from the ordered pairs (k,n*k-k^2) corresponding to integer points along the left side of the parabola b_k = n*k-k^2 where k is an odd integer such that 0 < k < floor(n/2).

Sum of the areas of the trapezoids with bases n and n-2i and height i for odd i in 0 <= i <= floor((n-1)/2). For a(n) the area formula for a trapezoid becomes (n+n-2i)*i/2 = (2n-2i)*i/2 = i*(n-i). For n=8, i=1,3 so a(8) = 1*(8-1) + 3*(8-3) = 7 + 15 = 22. - Wesley Ivan Hurt, Mar 21 2018

Sum of the areas of the symmetric L-shaped polygons with long side n/2 and odd width i in 0 <= i <= floor((n-1)/2). The area of each polygon is given by i^2+2i(n/2-i) = i^2+ni-2i^2 = i(n-i). For n=7, i=1,3 so 1(7-1) + 3(7-3) = 6 + 12 = 18. - 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((n-1)/2)} i * (n-i) * (i mod 2).

Conjectures from Colin Barker, Nov 20 2017: (Start)

G.f.: x^3*(2 + x + x^2 + x^3 + 7*x^4 + x^5 + x^6 + x^7 + x^8) / ((1 - x)^4*(1 + x)^3*(1 + x^2)^3).

a(n) = a(n-1) + 3*a(n-4) - 3*a(n-5) - 3*a(n-8) + 3*a(n-9) + a(n-12) - a(n-13) for n>13.

(End)

a(n) = (1/384)*(-1)^((-(-1)^n)/4)*((-2+2*(-1)^n)*((-1)^((4*n+2-(-1)^n)/4)-6*(-1)^(1/4)*I^n+(-1)^((2-(-1)^n)/4))+4*n*(6*n*(-1)^(1/4)*I^n+(-1)^((-1)^n/4)*(8-3*n*(1+(-1)^n)+4*n^2))) where I=sqrt(-1). - Wesley Ivan Hurt, Dec 02 2017

EXAMPLE

For n=8, the partitions are 3 + 5 and 1 + 7, so a(8) = 3*5 + 1*7 = 22.

MAPLE

A295317:=n->add(i*(n-i)*(i mod 2), i=1..floor((n-1)/2)): seq(A295317(n), n=1..100);

MATHEMATICA

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

PROG

(PARI) a(n) = sum(i=1, floor((n-1)/2), (i%2)*i*(n-i)) \\ Michael B. Porter, Dec 05 2017

CROSSREFS

Cf. A295318.

Sequence in context: A317944 A145520 A061412 * A165646 A261639 A024635

Adjacent sequences:  A295314 A295315 A295316 * A295318 A295319 A295320

KEYWORD

nonn,easy

AUTHOR

Wesley Ivan Hurt, Nov 19 2017

STATUS

approved

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Last modified October 15 09:40 EDT 2018. Contains 316211 sequences. (Running on oeis4.)