This site is supported by donations to The OEIS Foundation.

The October issue of the Notices of the Amer. Math. Soc. has an article about the OEIS.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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) 4

%I M4719 N2017

%S 1,0,0,0,10,60,462,3920,36954,382740,4327510,53088888,702756210,

%T 9988248956,151751644590,2454798429600,42130249479562,764681923900260,

%U 14636063499474054,294639009867223880

%N 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.

%C Number of directed Hamiltonian paths in the complement of C_n where C_n is the n-cycle graph. - _Andrew Howroyd_, Mar 15 2016

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

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

%H Vincenzo Librandi, <a href="/A002493/b002493.txt">Table of n, a(n) for n = 1..100</a>

%H M. Abramson and W. O. J. Moser, <a href="http://dx.doi.org/10.1214/aoms/1177698793">Permutations without rising or falling w-sequences</a>, Ann. Math. Stat., 38 (1967), 1245-1254.

%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013, p. 627-8.

%H A. J. Schwenk, <a href="/A002464/a002464_1.pdf">Letter to N. J. A. Sloane, Mar 24 1980</a> (with enclosure and notes)

%F 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

%F a(n) = Sum((-1)^(n-k)*k!*A102413(n,k),k=1..n), n>2. - _Vladeta Jovovic_, Nov 23 2007

%F 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

%F 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

%F 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

%p 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

%t 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 *)

%Y Cf. A002464, A002816, A006184.

%K nonn

%O 1,5

%A _N. J. A. Sloane_.

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 24 20:09 EDT 2018. Contains 315356 sequences. (Running on oeis4.)