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A307684
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Total volume of all rectangular prisms with dimensions r X s X t where r, s and t are the smallest, middle and largest parts in each partition of n into 3 parts.
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0
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0, 0, 1, 2, 7, 18, 34, 62, 121, 182, 292, 460, 651, 924, 1339, 1764, 2376, 3196, 4074, 5214, 6740, 8294, 10318, 12850, 15496, 18746, 22811, 26936, 32011, 38096, 44376, 51816, 60737, 69768, 80478, 92954, 105735, 120498, 137697, 155078, 175168, 198086, 221452
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OFFSET
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1,4
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LINKS
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FORMULA
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a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} i * k * (n-i-k).
G.f.: x^3*(1 + 3*x + 7*x^2 + 15*x^3 + 23*x^4 + 21*x^5 + 18*x^6 + 14*x^7 + 6*x^8) / ((1 - x)^6*(1 + x)^3*(1 + x + x^2)^4).
a(n) = -a(n-1) + 2*a(n-2) + 6*a(n-3) + 3*a(n-4) - 9*a(n-5) - 14*a(n-6) - 2*a(n-7) + 16*a(n-8) + 16*a(n-9) - 2*a(n-10) - 14*a(n-11) - 9*a(n-12) + 3*a(n-13) + 6*a(n-14) + 2*a(n-15) - a(n-16) - a(n-17) for n>17.
(End)
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EXAMPLE
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Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
1+1+8
1+1+7 1+2+7
1+2+6 1+3+6
1+1+6 1+3+5 1+4+5
1+1+5 1+2+5 1+4+4 2+2+6
1+1+4 1+2+4 1+3+4 2+2+5 2+3+5
1+1+3 1+2+3 1+3+3 2+2+4 2+3+4 2+4+4
1+1+1 1+1+2 1+2+2 2+2+2 2+2+3 2+3+3 3+3+3 3+3+4 ...
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n | 3 4 5 6 7 8 9 10 ...
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a(n) | 1 2 7 18 34 62 121 182 ...
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MATHEMATICA
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Table[Sum[Sum[i*k*(n - i - k), {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 0, 100}]
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PROG
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(PARI) a(n) = sum(k=1, n\3, sum(i=k, (n-k)\2, i*k*(n-i-k))); \\ Michel Marcus, Apr 22 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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