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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] (* From 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: A318681 A183207 A304797 * A253608 A126977 A102237

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 October 14 02:29 EDT 2019. Contains 327995 sequences. (Running on oeis4.)