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 A178312 a(n) = n * T(ceiling(n/2)), where T are the triangular numbers, A000217. 1
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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 LINKS Harvey P. Dale, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1). FORMULA a(n) = (n, n, n, n, n, ...) * (1, 0, 2, 0, 3, ...) using length n vectors. a(n) = n * A000217(A110654(n)). 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 a(n) = n*ceiling(n/2)*(ceiling(n/2)+1)/2 = (n+1)*A008805(n). - Wesley Ivan Hurt, Sep 21 2013 EXAMPLE 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 MAPLE a:= proc(n) local m; m:= ceil (n/2); n *m *(m+1) /2 end: seq (a(n), n=0..60); seq(k*sum(i, i=1..ceil(k/2), k=0..70)); # Wesley Ivan Hurt, Sep 21 2013 MATHEMATICA 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 *) PROG (Magma) [n*Ceiling(n/2)*(Ceiling(n/2)+1)/2: n in [0..44]]; // Bruno Berselli, Feb 14 2011 CROSSREFS Cf. A000217, A008805, A110654. Sequence in context: A304797 A337360 A340038 * A347495 A253608 A126977 Adjacent sequences: A178309 A178310 A178311 * A178313 A178314 A178315 KEYWORD nonn AUTHOR Gary W. Adamson, Dec 20 2010 EXTENSIONS Edited by Alois P. Heinz, Dec 23 2010 STATUS approved

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Last modified August 12 04:07 EDT 2024. Contains 375085 sequences. (Running on oeis4.)