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

0,3

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 408.

Formula

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

Mathematica

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+1<k, 0, Binomial[n-k+1, k]-Binomial[n-k-1, k-2]]);
semiwazt[k_, n_]:=(psum=0;
Do[p=IntegerPartitions[k, {size}];
psum=psum+Sum[permopak[p[[i]], k]*Binomial[n, Length[p[[i]]]]*Product[waz1t[p[[i, j]], n], {j, 1, Length[p[[i]]]}], {i, 1, Length[p]}], {size, 1, n}]; psum);
Table[semiwazt[n, n], {n, 1, 25}]
Join[{1}, Table[n Binomial[n^2-n-1, n-1]+n (-1)^n, {n, 2, 20}]] (* Harvey P. Dale, Nov 24 2016 *)

Prog

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

Crossrefs

Cf. A067961, A197989.

Sequence in context: A133018 A210343 A104168 * A362838 A068327 A066842

Adjacent sequences: A197987 A197988 A197989 * A197991 A197992 A197993

Keyword

nonn,nice

Author

Vaclav Kotesovec, Oct 20 2011

Extensions

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

Status

approved