A004652 - OEIS

A004652

Expansion of x*(1+x^2+x^4)/((1-x)*(1-x^2)*(1-x^3)).

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

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OFFSET

0,4

COMMENTS

As a Molien series this arises as (1+x^12)/((1-x^4)*(1-x^8)^2).

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

a(n) is also the number of inequivalent (modulo C_4 rotations) square n X n grids with squares coming in two colors and one square has one of the colors. See the formula from A054772. - Wolfdieter Lang, Oct 03 2016

Also the genus of the complete bipartite graph K_{n+2,n+2}. - Eric W. Weisstein, Jan 19 2018

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

A. R. Calderbank and N. J. A. Sloane, Double circulant codes over Z_4, J. Algeb. Combin., 6 (1997) 119-131 (Abstract, pdf, ps).

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

J. E. Strapasson, S. I. R. Costa, and M. M. S. Alves, On Genus of Circulant Graphs, arXiv:1004.0244 [math.GN], 2010-2016. - Jonathan Vos Post, Apr 05 2010

Eric Weisstein's World of Mathematics, Complete Bipartite Graph

Eric Weisstein's World of Mathematics, Graph Genus

Index entries for Molien series

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

FORMULA

a(n) = 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, 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). - Richard Choulet, Nov 27 2008

a(n) = n^2/4 - 3*((-1)^n-1)/8. - Omar E. Pol, Sep 28 2011

a(n) = -n + floor( (n+1)(n+3)/4 ). - Wesley Ivan Hurt, Jun 23 2013

a(n) = A054772(n, 1) = A054772(n, n^2-1), n >= 1. - Wolfdieter Lang, Oct 03 2016

E.g.f.: (x*(x + 1)*exp(x) + 3*sinh(x))/4. - Ilya Gutkovskiy, Oct 03 2016

a(n) = binomial(floor((n+3)/2),2) + binomial(floor((n+(-1)^n)/2),2). - Yuchun Ji, Feb 03 2021

EXAMPLE

From Gary W. Adamson, May 14 2010: (Start)

First few rows of the generating triangle =

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)

x + x^2 + 3*x^3 + 4*x^4 + 7*x^5 + 9*x^6 + 13*x^7 + 16*x^8 + 21*x^9 + ...

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+3), m=0..57) ; # Zerinvary Lajos, Mar 09 2007

MATHEMATICA

CoefficientList[Series[x (1 - x + x^2)/((1 - x)^2*(1 - x^2)), {x, 0, 57}], x] (* Michael De Vlieger, Oct 03 2016 *)

Table[Ceiling[n^2/4], {n, 0, 20}] (* Eric W. Weisstein, Jan 19 2018 *)

Ceiling[Range[0, 20]^2/4] (* Eric W. Weisstein, Jan 19 2018 *)

LinearRecurrence[{2, 0, -2, 1}, {1, 1, 3, 4}, {0, 20}] (* Eric W. Weisstein, Jan 19 2018 *)

PROG

(PARI) {a(n) = ceil(n^2 / 4)}

(Magma) [Ceiling(n^2/4): n in [0..60] ]; // Vincenzo Librandi, Aug 19 2011

(Haskell)

a004652 = ceiling . (/ 4) . fromIntegral . (^ 2)

a004652_list = 0 : 1 : zipWith (+) a004652_list [1..]

-- Reinhard Zumkeller, Dec 18 2013