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A110660
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Oblong (promic) numbers repeated.
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15
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0, 0, 2, 2, 6, 6, 12, 12, 20, 20, 30, 30, 42, 42, 56, 56, 72, 72, 90, 90, 110, 110, 132, 132, 156, 156, 182, 182, 210, 210, 240, 240, 272, 272, 306, 306, 342, 342, 380, 380, 420, 420, 462, 462, 506, 506, 552, 552, 600, 600, 650, 650, 702, 702, 756, 756, 812, 812
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OFFSET
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0,3
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COMMENTS
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Sum of the even numbers among the smallest parts in the partitions of 2n into two parts (see example). - Wesley Ivan Hurt, Jul 25 2014
For n > 0, a(n-1) is the sum of the smallest parts of the partitions of 2n into two distinct even parts. - Wesley Ivan Hurt, Dec 06 2017
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LINKS
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FORMULA
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a(n) = floor(n/2) * (floor(n/2)+1).
G.f.: 2*x^2/((1-x)^3*(1+x)^2);
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), for n > 4;
a(n) = (2*n^2 + 2*n - 1 + (2*n + 1)*(-1)^n)/8. (End)
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EXAMPLE
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a(4) = 6; The partitions of 2*4 = 8 into two parts are: (7,1), (6,2), (5,3), (4,4). The sum of the even numbers from the smallest parts of these partitions gives: 2 + 4 = 6.
a(5) = 6; The partitions of 2*5 = 10 into two parts are: (9,1), (8,2), (7,3), (6,4), (5,5). The sum of the even numbers from the smallest parts of these partitions gives: 2 + 4 = 6.
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MAPLE
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MATHEMATICA
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Table[Floor[n/2] (Floor[n/2] + 1), {n, 0, 50}] (* Wesley Ivan Hurt, Jul 25 2014 *)
CoefficientList[Series[2*x^2/((1 - x)^3*(1 + x)^2), {x, 0, 50}], x] (* Wesley Ivan Hurt, Jul 25 2014 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 0, 2, 2, 6}, 60] (* Harvey P. Dale, Jan 23 2021 *)
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PROG
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(Magma) k:=1; f:=func<n | n*(k*n+1)>; [0] cat [f(n*m): m in [-1, 1], n in [1..30]]; // Bruno Berselli, Nov 14 2012
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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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EXTENSIONS
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STATUS
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approved
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