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A295320
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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 larger part odd.
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2
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0, 0, 0, 3, 6, 5, 10, 22, 34, 30, 46, 73, 100, 91, 124, 172, 220, 204, 260, 335, 410, 385, 470, 578, 686, 650, 770, 917, 1064, 1015, 1176, 1368, 1560, 1496, 1704, 1947, 2190, 2109, 2370, 2670, 2970, 2870, 3190, 3553, 3916, 3795, 4180, 4612, 5044, 4900, 5356
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OFFSET
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1,4
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COMMENTS
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Sum of the areas of the distinct rectangles with odd length and integer 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 (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).
Sum of the areas of the trapezoids with bases n and n-2i and height i for odd values of n-i where i is 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=10, n-i is odd when i=1,3 so a(10) = 1*(10-1) + 3*(10-3) = 30. - Wesley Ivan Hurt, Mar 21 2018
Sum of the areas of the symmetric L-shaped polygons with long side n/2 and width i such that n-i is odd for 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=8, 8-i is odd for i=1,3 so 1(8-1) + 3(8-3) = 7 + 15 = 22. - 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) * ((n-i) mod 2).
G.f.: x^4*(3+3*x-x^2+5*x^3+3*x^4+3*x^5-x^6+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)*(-1)^(-(-1)^n/4)*((2-2*(-1)^n)*(-(-1)^((4*n+2-(-1)^n)/4)+6*(-1)^((6*n+1)/4)-(-1)^((2-(-1)^n)/4))+4*n*(12*(-1)^((-1)^n/4)*(-1)^n-6*n*(-1)^((2*n-1-2*(-1)^n)/4)+(-1)^((-1)^n/4)*(-4-3*n*(1+(-1)^n)+4*n^2))). - Wesley Ivan Hurt, Dec 03 2017
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EXAMPLE
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For n=8, the partitions into two distinct parts are 7 + 1, 6 + 2, and 5 + 3. Of these, 7 + 1 and 5 + 3 have the smaller part odd, so a(8) = 7*1 + 5*3 = 22. - Michael B. Porter, Dec 05 2017
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MAPLE
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A295320:=n->add(i*(n-i)*((n-i) mod 2), i=1..floor((n-1)/2)): seq(A295320(n), n=1..100);
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MATHEMATICA
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Table[Sum[i (n - i) Mod[n - i, 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*(n-i)*((n-i) % 2)); \\ Michel Marcus, Mar 26 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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