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 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
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 =
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)
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
CROSSREFS
Column 1 of A195040. - Omar E. Pol, Sep 28 2011
Cf. A054772, column 2.
KEYWORD
nonn,easy
AUTHOR
STATUS
approved