A008794 Squares repeated; a(n) = floor(n/2)^2.

0, 0, 1, 1, 4, 4, 9, 9, 16, 16, 25, 25, 36, 36, 49, 49, 64, 64, 81, 81, 100, 100, 121, 121, 144, 144, 169, 169, 196, 196, 225, 225, 256, 256, 289, 289, 324, 324, 361, 361, 400, 400, 441, 441, 484, 484, 529, 529, 576, 576

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also number of non-attacking kings on (n-1) X (n-1) board (cf. A030978). - Koksal Karakus (karakusk(AT)hotmail.com), May 27 2002

Also the independence number and clique covering number of the (n-1) X (n-1) king graph. - Eric W. Weisstein, Jun 20 2017

Maximum number of 2 X 2 tiles that fit on an n X n board. - Jon Perry, Aug 10 2003

(n)-(1) + (n-1)-(2) + (n-3)-(3) + ... + (n-r)-(r) ... n terms. E.g., 5-1+4-2+3 = 9, 6-1+5-2+4-3 = 9, 7-1+6-2+5-3+4 = 16, 8-1+7-2+6-3+5-4 = 16. - Amarnath Murthy, Jul 24 2005

The smallest possible number of white cells in a solution to an n X n nurikabe grid. - Tanya Khovanova, Feb 24 2009

(1 + x + 4*x^2 + 4*x^3 + 9*x^4 + ...) = (1/(1-x))*(1 + 3*x^2 + 5*x^4 + 7*x^6 + ...). - Gary W. Adamson, Apr 07 2010

If the set {1,2,...,n} is divided in half (a part having size ceiling(n/2) and the rest), then a(n+1) is the largest possible difference between the totals of these parts. - Vladimir Shevelev, Oct 14 2017

a(n+1) is the sum of the smallest parts of the partitions of 2n into two odd parts. - Wesley Ivan Hurt, Dec 06 2017

a(n-1) is the largest number of single cells of an n X n grid that share no edge or vertex with each other or those of the grid perimeter. - Stefano Spezia, Jul 30 2021

The binomial transform is 0, 0, 1, 4, 14, 44, 128, 352, 928, 2368, 5888... (see A007466). - R. J. Mathar, Feb 25 2023

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Stefano Spezia, Illustration of initial terms
• Eric Weisstein's World of Mathematics, Clique Covering Number.
• Eric Weisstein's World of Mathematics, King Graph.
• Eric Weisstein's World of Mathematics, Kings Problem.
• Index entries for sequences related to partitions
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A030978, A032766, A059841, A086832, A109613, A182579, A189889.

Programs

GAP

Flat(List([0..24],n->[n^2,n^2])); # Muniru A Asiru, Oct 09 2018

Magma

[(2*n-1)*(-1)^n/8+(2*n^2-2*n +1)/8: n in [0..60]]; // Vincenzo Librandi, Aug 21 2011

Maple

A008794:=n->floor(n/2)^2: seq(A008794(n), n=0..50); # Wesley Ivan Hurt, Dec 08 2017

Mathematica

With[{sq = Range[0, 30]^2}, Riffle[sq, sq]] (* Harvey P. Dale, Nov 20 2015 *)
Table[Floor[n/2]^2, {n, 0, 49}] (* Michael De Vlieger, Oct 21 2016 *)
Table[(2 n - 1) (-1)^n/8 + (2 n^2 - 2 n + 1)/8, {n, 0, 49}] (* Michael De Vlieger, Oct 21 2016 *)
CoefficientList[Series[x^2*(1 + x^2)/((1 - x) (1 - x^2)^2), {x, 0, 49}], x] (* Michael De Vlieger, Oct 21 2016 *)
CoefficientList[Series[((x^2-x)Cosh[x]+(1+x+x^2)Sinh[x])/4,{x,0,50}],x]*Table[k!,{k,0,50}] (* Stefano Spezia, Oct 07 2018 *)

PARI

a(n)=(n\2)^2 \\ Charles R Greathouse IV, Sep 24 2015

PARI

first(n) = Vec(x^2*(1 + x^2)/((1 - x)*(1 - x^2)^2) + O(x^n), -n) \\ Iain Fox, Dec 08 2017

Python

def A008794(n): return (n//2)**2 # Chai Wah Wu, Jun 07 2022

Sage

[((-1)^n*(2*n-1) +(2*n^2-2*n +1))/8 for n in (0..50)] # G. C. Greubel, Sep 11 2019

Formula

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

a(n) = floor(n/2)^2.

From Paul Barry, May 31 2003: (Start)

a(n) = (2*n - 1)*(-1)^n/8 + (2*n^2 - 2*n + 1)/8.

a(n+1) = Sum_{k=0..n} k*(1-(-1)^k)/2. (End)

a(n+2) = Sum_{k=0..n} A109613(k)*A059841(n-k). - Reinhard Zumkeller, Dec 05 2009

a(n) = A182579(n,n-2) for n > 1. - Reinhard Zumkeller, May 07 2012

3*a(n) = A032766(n)^2 - A032766(n^2). - Bruno Berselli, Oct 21 2016

a(n) = Sum_{i=1..n-1; i odd} i. - Olivier Pirson, Nov 06 2017

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), n > 4. - Iain Fox, Dec 08 2017

E.g.f.: ((x^2 - x)*cosh(x) + (1 + x + x^2)*sinh(x))/4. - Stefano Spezia, Oct 07 2018