A197989 -id:A197989 - cp's OEIS frontend

A197990 Number of binary arrangements of total n 1's, without adjacent 1's on n X n torus connected n-s.

1, 1, 4, 27, 664, 19375, 712536, 31474709, 1623421808, 95752130751, 6356272757680, 468976366239799, 38071162011854412, 3372179632719015287, 323631920261745650114, 33452466695808298399785, 3705187274710433648959456, 437779689881887196512539391

Offset: 0

Vaclav Kotesovec, Oct 20 2011

Andrew Howroyd, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 408.

Cf. A067961, A197989.

permopak[part_,k_]:=(hist=ConstantArray[0,k];
Do[hist[[part[[t]]]]++,{t,1,Length[part]}];
(Length[part])!/Product[(hist[[t]])!,{t,1,k}]);
waz1t[k_,n_]:=(If[n-k+1Harvey P. Dale, Nov 24 2016 *)

a(n) = if(n<=1, 1, n*binomial(n^2-n-1,n-1) + n*(-1)^n) \\ Andrew Howroyd, Mar 27 2023

a(n) = n*binomial(n^2-n-1,n-1) + n*(-1)^n, n > 1. - Vaclav Kotesovec, Oct 20 2011

a(0)=1 prepended by Andrew Howroyd, Mar 27 2023

A182563 Number of ways to place n non-attacking semi-knights on an n x n chessboard.

Original entry on oeis.org

1, 6, 70, 1289, 33864, 1148760, 47700972, 2344465830, 133055587660, 8559364525414, 615266768106190, 48861588247978827, 4247584874013608724, 401107335066453376830, 40880928693752664368224, 4472281486633326131737868

Offset: 1

Vaclav Kotesovec, May 05 2012

Semi-knight is a semi-leaper [1,2]. Moves of a semi-knight are allowed only in [2,1] and [-2,-1]. See also semi-bishops (A187235).

V. Kotesovec, Non-attacking chess pieces, 6ed, p.289

Cf. A182562, A201540, A201513, A002465, A201511, A197989, A201861.

Asymptotic: a(n) ~ n^(2n)/n!*exp(-3/2).

a(16) from Vaclav Kotesovec, May 24 2021

A244288 Number of binary arrangements of total n 1's, without adjacent 1's on n X n array connected nw-se.

Original entry on oeis.org

1, 1, 5, 57, 1084, 29003, 999717, 42125233, 2096106904, 120194547233, 7799803041491, 564856080384900, 45146219773912540, 3946445378386791157, 374482268128153003615, 38330653031858936914329, 4209191997519328986666624, 493575737047609363968826907

Offset: 0

Vaclav Kotesovec, Jun 25 2014

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.422

Cf. A197989, A201540, A201511, A201861, A201513, A244284.

P(m,n) = sum(k=0, (m+1)\2, binomial(m-k+1,k)*x^k, O(x*x^n))
a(n) = polcoef(P(n,n)*prod(m=1, n-1, P(m,n))^2, n) \\ Andrew Howroyd, Mar 27 2023

a(n) ~ n^(2*n)/n! * exp(-3/2).

a(16) from Vaclav Kotesovec, Sep 04 2016
a(17) from Vaclav Kotesovec, Jun 15 2021
a(0)=1 prepended by Andrew Howroyd, Mar 27 2023

cp's OEIS Frontend

A197990 Number of binary arrangements of total n 1's, without adjacent 1's on n X n torus connected n-s.

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A182563 Number of ways to place n non-attacking semi-knights on an n x n chessboard.

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A244288 Number of binary arrangements of total n 1's, without adjacent 1's on n X n array connected nw-se.

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PARI

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Extensions