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A296141
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Sum of the smaller parts of the partitions of n into two distinct parts with the larger part even.
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0
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0, 0, 1, 0, 1, 2, 4, 2, 4, 6, 9, 6, 9, 12, 16, 12, 16, 20, 25, 20, 25, 30, 36, 30, 36, 42, 49, 42, 49, 56, 64, 56, 64, 72, 81, 72, 81, 90, 100, 90, 100, 110, 121, 110, 121, 132, 144, 132, 144, 156, 169, 156, 169, 182, 196, 182, 196, 210, 225, 210, 225, 240
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OFFSET
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1,6
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COMMENTS
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a(n+1) is the sum of the smaller parts in the partitions of n into two parts with the larger part odd. For example, a(11) = 9; the partitions of 10 into two parts are (9,1), (8,2), (7,3), (6,4) and (5,5). Three of these partitions have an odd number as their larger part, namely (9,1), (7,3) and (5,5). Adding the smaller parts of these partitions gives 1 + 3 + 5 = 9.
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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+1) mod 2).
G.f.: x^3*(1 - x + x^2 + x^3) / ((1 - x)^3*(1 + x)^2*(1 + x^2)^2).
a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9) for n > 9.
(End)
a(n) = floor((n+1)/4)^2*(n mod 2)+(1+floor((n-2)/4))*floor((n-2)/4)*((n+1) mod 2). - Wesley Ivan Hurt, Dec 08 2017
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EXAMPLE
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a(10) = 6; the partitions of 10 into two parts are (9,1), (8,2), (7,3), (6,4) and (5,5). Two of these partitions have an even number as their larger part, namely (8,2) and (6,4). Adding the smaller parts of these partitions gives 2 + 4 = 6.
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MATHEMATICA
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Table[Sum[i Mod[n - 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, floor((n-1)/2), i*lift(Mod(n-i+1, 2))) \\ Iain Fox, Dec 06 2017
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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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