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A178312
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a(n) = n * T(ceiling(n/2)), where T are the triangular numbers, A000217.
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1
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0, 1, 2, 9, 12, 30, 36, 70, 80, 135, 150, 231, 252, 364, 392, 540, 576, 765, 810, 1045, 1100, 1386, 1452, 1794, 1872, 2275, 2366, 2835, 2940, 3480, 3600, 4216, 4352, 5049, 5202, 5985, 6156, 7030, 7220, 8190, 8400, 9471, 9702, 10879, 11132
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OFFSET
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0,3
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COMMENTS
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This is n times the sum of the second parts of the partitions of n+1 into exactly two parts. - Wesley Ivan Hurt, Sep 21 2013
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LINKS
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FORMULA
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a(n) = (n, n, n, n, n, ...) * (1, 0, 2, 0, 3, ...) using length n vectors.
G.f.: x*(1+x+4*x^2)/((1+x)^3*(1-x)^4).
a(n) = n*(2*n*(n+3) - (2*n+3)*(-1)^n + 3)/16. - Bruno Berselli, Feb 14 2011
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EXAMPLE
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a(4) = 12 = (4, 4, 4, 4) * (1, 0, 2, 0) = (4 + 0 + 8 + 0).
a(5) = 30; 5+1=6 has exactly 3 partitions into 2 parts: (5,1),(4,2),(3,3). Adding the second parts of these partitions and multiplying the result by 5, we get 5*6=30. - Wesley Ivan Hurt, Sep 21 2013
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MAPLE
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a:= proc(n) local m; m:= ceil (n/2); n *m *(m+1) /2 end:
seq (a(n), n=0..60);
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MATHEMATICA
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nn=50; With[{tnos=Accumulate[Range[nn]]}, Table[n tnos[[Ceiling[n/2]]], {n, 0, nn}]] (* or *) LinearRecurrence[{1, 3, -3, -3, 3, 1, -1}, {0, 1, 2, 9, 12, 30, 36}, 50] (* Harvey P. Dale, May 12 2012 *)
Table[n*Ceiling[n/2]*(Ceiling[n/2]+1)/2, {n, 0, 100}] (* Wesley Ivan Hurt, Sep 21 2013 *)
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PROG
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(Magma) [n*Ceiling(n/2)*(Ceiling(n/2)+1)/2: n in [0..44]]; // Bruno Berselli, Feb 14 2011
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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