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A295292
Sum of the products of the smaller and larger parts of the partitions of n into two parts with the larger part odd.
2
0, 1, 0, 3, 6, 14, 10, 22, 34, 55, 46, 73, 100, 140, 124, 172, 220, 285, 260, 335, 410, 506, 470, 578, 686, 819, 770, 917, 1064, 1240, 1176, 1368, 1560, 1785, 1704, 1947, 2190, 2470, 2370, 2670, 2970, 3311, 3190, 3553, 3916, 4324, 4180, 4612, 5044, 5525
OFFSET
1,4
COMMENTS
Sum of the areas of the distinct rectangles with odd length and integer width such that L + W = n, W <= L. For example, a(8) = 22; the rectangles are 1 X 7 and 3 X 5, so 1*7 + 3*5 = 22.
Sum of the ordinates from the ordered pairs (n-k,n*k-k^2) corresponding to integer points along the right side of the parabola b_k = n*k-k^2 where n-k is an odd integer such that 0 < k <= floor(n/2).
FORMULA
a(n) = Sum_{i=1..floor(n/2)} i * (n - i) * ((n-i) mod 2).
Conjectures from Colin Barker, Nov 20 2017: (Start)
G.f.: x^2*(1 - x + 3*x^2 + 3*x^3 + 5*x^4 - x^5 + 3*x^6 + 3*x^7) / ((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)*((2-2*(-1)^n)*(1+(-1)^n+6*(-1)^((2*n-3-(-1)^n)/4))+16*n*(-1+3*(-1)^n)+12*n^2*(1+(-1)^n+2*(-1)^((1/4)*(3+(-1)^n))*(-1)^(3*n/2))+16*n^3). - Wesley Ivan Hurt, Dec 03 2017
EXAMPLE
a(8) = 22; the partitions of 8 into 2 parts are (7,1), (6,2), (5,3) and (4,4). Two of these partitions have odd numbers as their larger part: (7,1) and (5,3), so 7*1 + 5*3 = 22.
MAPLE
A295292:=n->add(i*(n-i)*((n-i) mod 2), i=1..floor(n/2)): seq(A295292(n), n=1..100);
MATHEMATICA
Table[Sum[i (n - i) Mod[n - i, 2], {i, Floor[n/2]}], {n, 80}]
CROSSREFS
Cf. A295293.
Sequence in context: A301656 A359669 A322183 * A053365 A101034 A342103
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Nov 19 2017
STATUS
approved