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A295318
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Sum of the products of the smaller and larger parts of the partitions of n into two distinct parts with the smaller part even.
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2
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0, 0, 0, 0, 6, 8, 10, 12, 34, 40, 46, 52, 100, 112, 124, 136, 220, 240, 260, 280, 410, 440, 470, 500, 686, 728, 770, 812, 1064, 1120, 1176, 1232, 1560, 1632, 1704, 1776, 2190, 2280, 2370, 2460, 2970, 3080, 3190, 3300, 3916, 4048, 4180, 4312, 5044, 5200, 5356
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OFFSET
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1,5
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COMMENTS
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Sum of the areas of the distinct rectangles with integer length and even 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 (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 even 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 even 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=9, i=0,2,4 so a(9) = 0*(9-0) + 2*(9-2) + 4*(9-4) = 0 + 14 + 20 = 34. - Wesley Ivan Hurt, Mar 22 2018
Sum of the areas of the symmetric L-shaped polygons with long side n/2 and even 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=9, i=0,2,4 so 0(9-0) + 2(9-2) + 4(9-4) = 0 + 14 + 20 = 34. - Wesley Ivan Hurt, Mar 26 2018
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LINKS
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FORMULA
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a(n) = Sum_{i=1..floor((n-1)/2)} i * (n-i) * ((i+1) mod 2).
G.f.: 2*x^5*(3 + x + x^2 + x^3 + 2*x^4) / ((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)^((2*n+1)/4)+(-1)^((2-(-1)^n)/4))+4*n*(-6*n*(-1)^((2*n+1)/4)+(-1)^((-1)^n/4)*(-16-3*n*(1+(-1)^n)+4*n^2))). - Wesley Ivan Hurt, Dec 02 2017
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EXAMPLE
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For n=9, the partitions are 7 + 2 and 5 + 4, so a(9) = 7*2 + 5*4 = 34. - Michael B. Porter, Dec 05 2017
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MAPLE
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A295318:=n->add(i*(n-i)*((i+1) mod 2), i=1..floor((n-1)/2)): seq(A295318(n), n=1..100);
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MATHEMATICA
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Table[Sum[i (n - i) Mod[i + 1, 2], {i, Floor[(n - 1)/2]}], {n, 80}]
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PROG
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(PARI) a(n) = sum(i=1, (n-1)\2, i*((i+1)%2)*(n-i)); \\ Altug Alkan, Mar 22 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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