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A004652
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Expansion of x*(1+x^2+x^4)/((1-x)*(1-x^2)*(1-x^3)).
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14
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0, 1, 1, 3, 4, 7, 9, 13, 16, 21, 25, 31, 36, 43, 49, 57, 64, 73, 81, 91, 100, 111, 121, 133, 144, 157, 169, 183, 196, 211, 225, 241, 256, 273, 289, 307, 324, 343, 361, 381, 400, 421, 441, 463, 484, 507, 529, 553, 576, 601, 625, 651, 676, 703, 729, 757, 784, 813
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| As a Molien series this arises as (1+x^12)/((1-x^4)*(1-x^8)^2).
1.1 on p.3 of Strapasson gives a graph theory use about circulant graphs of genus 1 of the formula a(n) = ceiling(n^2/4). The subsequence of primes in this sequence begins: 3, 7, 13, 31, 43, 73, 157, 211, 241, 307, 421, 463, 601, 757, 1123, 1483. [From Jonathan Vos Post, Apr 05 2010]
Starting (1, 3, 4,...) = row sums of an infinite triangle with alternate columns of (1, 2, 3,...) and (1, 0, 0, 0,...). [Gary W. Adamson, May 14 2010]
A000290 and A002061 (but without the first term of A002061) interleaved. - Omar E. Pol, Sep 28 2011
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..10000
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
A. R. Calderbank and N. J. A. Sloane, Double circulant codes over Z_4, J. Algeb. Combin., 6 (1997) 119-131 (Abstract, pdf, ps).
J. E. Strapasson, S. I. R. Costa, M. M. S. Alves, On Genus of Circulant Graphs, April 01, 2010. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Apr 05 2010]
Index entries for Molien series
Index to sequences with linear recurrences with constant coefficients, signature (2,0,-2,1).
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FORMULA
| ceiling(n^2/4).
a(-n) = a(n).
G.f.: x * (1 - x + x^2) / ((1 - x)^2 * (1 - x^2)).
a(n) = a(n-1) + a(n-2) - a(n-3) + 1. a(2*n) = n^2, a(2*n-1) = n^2 - n + 1 - Michael Somos, Apr 21, 2000.
Interleaves square numbers with centered polygonal numbers: a(2*n)=A000290(n), a(2*n+1)=A002061(n+1). - Paul Barry, Mar 13 2003
For n>1: a(n) is the digit reversal of n in base A008619(n), where a(n) is written in base 10. - Naohiro Nomoto (pcmusume(AT)m11.alpha-net.ne.jp), Mar 15 2004
a(n)=a(n-2)+n-1 - Paul Barry, Jul 14 2004
Euler transform of length 6 sequence [ 1, 2, 1, 0, 0, -1]. - Michael Somos Apr 03 2007
Starting (1, 3, 4, 7, 9, 13,...), row sums of triangle A135840. - Gary W. Adamson, Dec 01 2007
a(n)=(3/8)*(-1)^(n+1)+(5/8)-(3/4)*(n+1)+(1/4)*(n+2)*(n+1) [From Richard Choulet (richardchoulet(AT)yahoo.fr), Nov 27 2008]
a(n) = n^2/4-3*((-1)^n-1)/8. - Omar E. Pol, Sep 28 2011
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EXAMPLE
| Contribution from Gary W. Adamson, May 14 2010: (Start)
First few rows of the generating triangle =
1;
2, 1;
3, 0, 1;
4, 0, 2, 1;
5, 0, 3, 0, 1;
6, 0, 4, 0, 2, 1;
7, 0, 5, 0, 3, 0, 1;
8, 0, 6, 0, 4, 0, 2, 1;
...
Example: a(7) = 13 = (6 + 0 + 4 + 0 + 2 + 1). (End)
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MAPLE
| with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r), right=Set(U, card<r), U=Sequence(Z, card>=2)}, unlabeled]: subs(r=1, stack): seq(count(subs(r=2, ZL), size=m), m=3..57) ; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 09 2007
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PROG
| (PARI) a(n)=ceil(n^2/4)
(MAGMA) [Ceiling(n^2/4): n in [0..60] ]; // Vincenzo Librandi, Aug 19 2011
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CROSSREFS
| First differences give A028242. Cf. A035104, A035106.
A002061(n)=a(2*n-1). A035104(n)=a(n+7)-12. A035106(n)=a(n+3)-1.
Cf. A135840.
Column 1 of A195040. - Omar E. Pol, Sep 28 2011
Sequence in context: A073273 A072441 A152032 * A061568 A146994 A103054
Adjacent sequences: A004649 A004650 A004651 * A004653 A004654 A004655
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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