A002493 Number of ways to arrange n non-attacking kings on an n X n board, with 2 sides identified to form a cylinder, with 1 in each row and column. (Formerly M4719 N2017)

1, 0, 0, 0, 10, 60, 462, 3920, 36954, 382740, 4327510, 53088888, 702756210, 9988248956, 151751644590, 2454798429600, 42130249479562, 764681923900260, 14636063499474054, 294639009867223880

Offset

1,5

Comments

Number of directed Hamiltonian paths in the complement of C_n where C_n is the n-cycle graph. - Andrew Howroyd, Mar 15 2016

Number of ways of arranging n consecutive integers in a circle such that no pair of adjacent integers differ by 1, rotations are distinct. - Graham Holmes, Sep 03 2020

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..100

M. Abramson and W. O. J. Moser, Permutations without rising or falling w-sequences, Ann. Math. Stat., 38 (1967), 1245-1254.

Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, Counting King Permutations on the Cylinder, arXiv:2001.02948 [math.CO], 2020.

Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 627-8.

A. J. Schwenk, Letter to N. J. A. Sloane, Mar 24 1980 (with enclosure and notes)

Formula

The linear recurrence operator annihilating this sequence is (N is the shift operator Na(n):=a(n + 1)) is - 3*(43*n + 197)*(n - 2)*(n + 1)/( - 1222 + 753*n + 349*n^2) - 5*(n - 1)*(44*n^2 + 477*n + 1222)/( - 1222 + 753*n + 349*n^2)*N + 2*(n + 1)*(239*n^2 + 873*n - 1232)/( - 1222 + 753*n + 349*n^2)*N^2 + 4*(394 - 259*n + 215*n^2 + 55*n^3)/( - 1222 + 753*n + 349*n^2)*N^3 - ( - 7342 + 3699*n + 2718*n^2 + 349*n^3)/( - 1222 + 753*n + 349*n^2)*N^4 + N^5. - Doron Zeilberger, Nov 14 2007

a(n) = Sum((-1)^(n-k)*k!*A102413(n,k),k=1..n), n>2. - Vladeta Jovovic, Nov 23 2007

a(n) = b(n+1) - 2*Sum_{k=0..floor(n/2)} b(n-2*k) for n>1, where b(n)=A002464(n) if n>0 else b(0)=0. - Vladeta Jovovic, Nov 24 2007

Asymptotic: a(n) ~ n!/e^2*(1 - 2/n - 2/n^2 - 4/(3n^3) + 8/(3n^4) + 326/(15n^5) + 4834/(45n^6) + 154258/(315n^7) + 232564/(105n^8) + ...). - Vaclav Kotesovec, Apr 06 2012

a(n) = n! + Sum_{i=1..n-1} ((-1)^i * (n-i-1)! * n * Sum_{j=0..i-1} (2^(j+1) * C(i-1,j) * C(n-i,j+1))), for n>=5. - Andrew Woods, Jan 08 2015

Maple

b1:= proc(n, r) local gu, x; if r=0 then RETURN(0): fi: gu := (x*diff(x*(1+x)/(1-x), x))* (x*(1 + x)/(1 - x))^(r-1); gu := taylor(gu, x = 0, n +1); coeff(gu, x, n ) end: b:=proc(n) local r: if n=1 then 1 elif n=2 then 0 else add((-1)^(n-r)*r!*b1(n, r), r=0..n): fi: end: # Doron Zeilberger, Nov 14 2007

Mathematica

b[n_]:=(If[n>0, n!+Sum[(-1)^r*(n-r)!*Sum[2^c*Binomial[r-1, c-1]*Binomial[n-r, c], {c, 1, r}], {r, 1, n-1}], 0]); Table[If[n>2, b[n]-2*Sum[b[n-1-2k], {k, 0, Floor[n/2]}], If[n==1, 1, 0]], {n, 1, 25}] (* Vaclav Kotesovec after Vladeta Jovovic, Apr 06 2012 *)

Crossrefs

Cf. A002464, A002816, A006184.

Right diagonal of A338838.

Sequence in context: A250575 A155633 A368525 * A054364 A004309 A281863

Adjacent sequences: A002490 A002491 A002492 * A002494 A002495 A002496

Keyword

nonn

Author

N. J. A. Sloane

Status

approved