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A295287
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Sum of the products of the smaller and larger parts of the partitions of n into two parts with the smaller part even.
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7
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0, 0, 0, 4, 6, 8, 10, 28, 34, 40, 46, 88, 100, 112, 124, 200, 220, 240, 260, 380, 410, 440, 470, 644, 686, 728, 770, 1008, 1064, 1120, 1176, 1488, 1560, 1632, 1704, 2100, 2190, 2280, 2370, 2860, 2970, 3080, 3190, 3784, 3916, 4048, 4180, 4888, 5044, 5200
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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 integer length and even width such that L + W = n, W <= L. For example, a(8) = 28; the rectangles are 2 X 6 and 4 X 4, so 2*6 + 4*4 = 28.
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).
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LINKS
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FORMULA
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a(n) = Sum_{i=1..floor(n/2)} i * (n - i) * ((i+1) mod 2).
G.f.: 2*x^4*(2 + x + x^2 + x^3 + 3*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) = (1/96)*(3*n^2*(1+(-1)^n)-16*n+4*n^3+3*I^n*(1+I)*(-1)^(-(-1)^n/4)*((1+2*n^2)*(-1)^n-1)/sqrt(2)) where I=sqrt(-1). - Wesley Ivan Hurt, Dec 02 2017
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EXAMPLE
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a(10) = 40; the partitions of 10 into two parts are (9,1), (8,2), (7,3), (6,4), (5,5). Two of these partitions have their smaller part even, namely (8,2) and (6,4). So, a(10) = 8*2 + 6*4 = 40.
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MAPLE
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A295287:=n->add(i*(n-i)*((i+1) mod 2), i=1..floor(n/2)): seq(A295287(n), n=1..100);
# Alternative:
for j from 0 to 3 do
F[j]:= expand(simplify(eval(sum(2*i*(4*k+j-2*i), i=1..k)), {k=(n-j)/4}))
od:
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MATHEMATICA
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Table[Sum[i (n - i) Mod[i + 1, 2], {i, Floor[n/2]}], {n, 80}]
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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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