A178312 a(n) = n * T(ceiling(n/2)), where T are the triangular numbers, A000217.

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

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