A004652 - OEIS

%I #111 Feb 16 2025 08:32:28

%S 0,1,1,3,4,7,9,13,16,21,25,31,36,43,49,57,64,73,81,91,100,111,121,133,

%T 144,157,169,183,196,211,225,241,256,273,289,307,324,343,361,381,400,

%U 421,441,463,484,507,529,553,576,601,625,651,676,703,729,757,784,813

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

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

%C 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

%C 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

%C Also the genus of the complete bipartite graph K_{n+2,n+2}. - _Eric W. Weisstein_, Jan 19 2018

%H Vincenzo Librandi, <a href="/A004652/b004652.txt">Table of n, a(n) for n = 0..10000</a>

%H A. R. Calderbank and N. J. A. Sloane, Double circulant codes over Z_4, J. Algeb. Combin., 6 (1997) 119-131 (<a href="http://neilsloane.com/doc/mckay.txt">Abstract</a>, <a href="http://neilsloane.com/doc/mckay.pdf">pdf</a>, <a href="http://neilsloane.com/doc/mckay.ps">ps</a>).

%H G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://neilsloane.com/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.

%H J. E. Strapasson, S. I. R. Costa, and M. M. S. Alves, <a href="http://arxiv.org/abs/1004.0244">On Genus of Circulant Graphs</a>, arXiv:1004.0244 [math.GN], 2010-2016. - _Jonathan Vos Post_, Apr 05 2010

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphGenus.html">Graph Genus</a>

%H <a href="/index/Mo#Molien">Index entries for Molien series</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).

%F a(n) = ceiling(n^2/4).

%F a(-n) = a(n).

%F G.f.: x * (1 - x + x^2) / ((1 - x)^2 * (1 - x^2)).

%F 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

%F Interleaves square numbers with centered polygonal numbers: a(2*n)=A000290(n), a(2*n+1)=A002061(n+1). - _Paul Barry_, Mar 13 2003

%F 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

%F a(n) = a(n-2) + n - 1. - _Paul Barry_, Jul 14 2004

%F Euler transform of length 6 sequence [ 1, 2, 1, 0, 0, -1]. - _Michael Somos_, Apr 03 2007

%F Starting (1, 3, 4, 7, 9, 13, ...), row sums of triangle A135840. - _Gary W. Adamson_, Dec 01 2007

%F a(n) = (3/8)*(-1)^(n+1) + 5/8 - (3/4)*(n+1) + (1/4)*(n+2)*(n+1). - _Richard Choulet_, Nov 27 2008

%F a(n) = n^2/4 - 3*((-1)^n-1)/8. - _Omar E. Pol_, Sep 28 2011

%F a(n) = -n + floor( (n+1)(n+3)/4 ). - _Wesley Ivan Hurt_, Jun 23 2013

%F a(n) = A054772(n, 1) = A054772(n, n^2-1), n >= 1. - _Wolfdieter Lang_, Oct 03 2016

%F E.g.f.: (x*(x + 1)*exp(x) + 3*sinh(x))/4. - _Ilya Gutkovskiy_, Oct 03 2016

%F a(n) = binomial(floor((n+3)/2),2) + binomial(floor((n+(-1)^n)/2),2). - _Yuchun Ji_, Feb 03 2021

%e From _Gary W. Adamson_, May 14 2010: (Start)

%e First few rows of the generating triangle =

%e 1;

%e 2, 1;

%e 3, 0, 1;

%e 4, 0, 2, 1;

%e 5, 0, 3, 0, 1;

%e 6, 0, 4, 0, 2, 1;

%e 7, 0, 5, 0, 3, 0, 1;

%e 8, 0, 6, 0, 4, 0, 2, 1;

%e ...

%e Example: a(7) = 13 = (6 + 0 + 4 + 0 + 2 + 1). (End)

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

%p 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

%t CoefficientList[Series[x (1 - x + x^2)/((1 - x)^2*(1 - x^2)), {x, 0, 57}], x] (* _Michael De Vlieger_, Oct 03 2016 *)

%t Table[Ceiling[n^2/4], {n, 0, 20}] (* _Eric W. Weisstein_, Jan 19 2018 *)

%t Ceiling[Range[0, 20]^2/4] (* _Eric W. Weisstein_, Jan 19 2018 *)

%t LinearRecurrence[{2, 0, -2, 1}, {1, 1, 3, 4}, {0, 20}] (* _Eric W. Weisstein_, Jan 19 2018 *)

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

%o (Magma) [Ceiling(n^2/4): n in [0..60] ]; // _Vincenzo Librandi_, Aug 19 2011

%o (Haskell)

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

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

%o -- _Reinhard Zumkeller_, Dec 18 2013

%Y First differences give A028242. Cf. A035104, A035106.

%Y A002061(n)=a(2*n-1). A035104(n)=a(n+7)-12. A035106(n)=a(n+3)-1.

%Y Cf. A135840, A000290.

%Y Column 1 of A195040. - _Omar E. Pol_, Sep 28 2011

%Y Cf. A054772, column 2.

%K nonn,easy

%O 0,4

%A _N. J. A. Sloane_