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A308025
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a(n) = n*(2*n - 3 - (-1)^n)*(5*n - 2 + (-1)^n)/16.
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1
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0, 0, 9, 19, 55, 87, 168, 234, 378, 490, 715, 885, 1209, 1449, 1890, 2212, 2788, 3204, 3933, 4455, 5355, 5995, 7084, 7854, 9150, 10062, 11583, 12649, 14413, 15645, 17670, 19080, 21384, 22984, 25585, 27387, 30303, 32319, 35568, 37810, 41410, 43890, 47859
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OFFSET
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1,3
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COMMENTS
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Consider the rectangular prisms with dimensions s X t X t, where n = s + t and s < t. Then a(n) is the sum of the areas of the squares that rest on a given space diagonal in each of the rectangular prisms.
Sum of the squares of the smaller parts and twice the sum of the squares of the larger parts in the partitions of n into two distinct parts.
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LINKS
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FORMULA
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a(n) = Sum_{i=1..floor((n-1)/2)} i^2 + 2*(n-i)^2.
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7).
G.f.: x^3*(9 + 10*x + 9*x^2 + 2*x^3) / ((1 - x)^4*(1 + x)^3). - Colin Barker, May 17 2019
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MATHEMATICA
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Table[n*(2 n - 3 - (-1)^n)*(5 n - 2 + (-1)^n)/16, {n, 60}]
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PROG
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(PARI) concat([0, 0], Vec(x^3*(9 + 10*x + 9*x^2 + 2*x^3) / ((1 - x)^4*(1 + x)^3) + O(x^40))) \\ Colin Barker, May 17 2019
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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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