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A008794 Squares repeated; a(n) = floor(n/2)^2. 37
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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

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).

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) = ( sqrt( Sum_{j=0..n} (j+1)*(cos(j*Pi) + 1)/2 ) - 1 )^2. - Paolo P. Lava, Dec 04 2006

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

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 *)

PROG

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

(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

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

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

(Python)

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

CROSSREFS

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

Sequence in context: A206919 A168039 A145445 * A075709 A332777 A238629

Adjacent sequences: A008791 A008792 A008793 * A008795 A008796 A008797

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 6 09:26 EST 2022. Contains 358608 sequences. (Running on oeis4.)