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A002493
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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)
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5
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1, 0, 0, 0, 10, 60, 462, 3920, 36954, 382740, 4327510, 53088888, 702756210, 9988248956, 151751644590, 2454798429600, 42130249479562, 764681923900260, 14636063499474054, 294639009867223880
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OFFSET
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1,5
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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FORMULA
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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) = 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
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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