A276356 -id:A276356 - cp's OEIS frontend

A269562 Array read by antidiagonals: T(n,m) is the number of Hamiltonian cycles in the rook graph K_n X K_m.

0, 0, 0, 1, 1, 1, 3, 3, 3, 3, 12, 30, 48, 30, 12, 60, 480, 1566, 1566, 480, 60, 360, 12000, 126120, 284112, 126120, 12000, 360, 2520, 430920, 18153720, 122330880, 122330880, 18153720, 430920, 2520, 20160, 21052080, 4357332000, 112777827840, 335750676480, 112777827840, 4357332000, 21052080, 20160

Offset: 1

Andrew Howroyd, Feb 29 2016

Equivalently, the number of rook tours on an n X m lattice.

2*T(n,m) is divisible by (n-1)!*(m-1)!. - Andrew Howroyd, Feb 08 2021

			Array begins:
=============================================================
n\m |   1      2          3            4                5
----+--------------------------------------------------------
  1 |   0      0          1            3               12 ...
  2 |   0      1          3           30              480 ...
  3 |   1      3         48         1566           126120 ...
  4 |   3     30       1566       284112        122330880 ...
  5 |  12    480     126120    122330880     335750676480 ...
  6 |  60  12000   18153720 112777827840 2190773906150400 ...
  7 | 360 430920 4357332000 ...
     ...

Andrew Howroyd, Table of n, a(n) for n = 1..96
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Eric Weisstein's World of Mathematics, Rook Graph

Column 1 is A001710(n-1) for n >= 3.
Columns 2..4 are A276356, A341498, A341499.
Main diagonal is A269561.
Cf. A269565, A286418.

From Andrew Howroyd, Feb 08 2021: (Start)
T(n,m) = T(m,n).
T(n,1) = (n-1)!/2 for n >= 3. (End)

A089039 Number of circular permutations of 2n letters that are free of jealousy.

Original entry on oeis.org

1, 2, 6, 60, 960, 24000, 861840, 42104160, 2686763520, 217039253760, 21651071904000, 2614084251609600, 375698806311628800, 63383303286471168000, 12403896267489382656000, 2786994829444848422400000, 712575504763406361133056000

Offset: 1

Akemi Nakamura, Michihiro Takahashi, Shogaku Meitantei (naka(AT)sansu.org), Dec 03 2003

The number of circular permutations of 2*n people consisting of n married couples, such that no one sits next to a person of the opposite sex who is not his or her spouse.

Limit_{n->oo} a(n)/(n-1)!^2 = Sum_{k>=1} 1/(k!*(k-1)!) = 1.590636854637329063382254424999666247954478159495536647132... (A096789).

			a(3)=6 because ABCcba, ACBbca, ABbacC, ACcabB, AabcCB, AacbBC are possible.

Eiji Kurihara, Small room of mathematics; see the answer for No. 380 of arithmetic challenges version 1.
Mikhail Kurkov and others, On a A089039 and pair of sequences with simple recursion, Math Overflow, Apr 20 2024
Masaru Yoshikawa, Arithmetic challenges. See problem No. 380.

a[1] = 1; a[n_] := n!*(n-2)!*HypergeometricPFQ[{1-n/2, 3/2-n/2}, {2, 2-n, 2-n}, 4]; Table[a[n], {n, 1, 17}] (* Jean-François Alcover, Oct 30 2013, after symbolic sum *)

a(n) = if (n==1, 1, sum (k=1, n\2, n!*(n-k-1)!^2/((k-1)!^2*(n-2*k)!*k))); \\ Michel Marcus, Sep 03 2013

a(1)=1, a(n) = Sum_{k=1..floor(n/2)} n!*(n-k-1)!^2/((k-1)!^2*(n-2*k)!*k) for n > 1.
a(n) = (n-1)!*(A001040(n-1) + A001053(n)) = 2*A276356(n), n > 1. - Conjectured by Mikhail Kurkov, Feb 10 2019 and proved by Max Alekseyev, Apr 23 2024 (see MO link)
a(n+4) = -(n+3)*(n+2)*(n*(n+1)*a(n) + 2*(n+1)^2*a(n+1) + n*(n+3)*a(n+2) - 2*a(n+3)) for all integer n>1. - conjectured by Michael Somos, Apr 21 2024. [The conjecture is equivalent to Kurkov's formula and thus is also proved. - Max Alekseyev, Apr 23 2024]

A360878 Number of (undirected) paths in the 2 X n rook graph.

Original entry on oeis.org

1, 12, 129, 1984, 45945, 1524156, 68838217, 4070403744, 305642504529, 28440008182540, 3214141725643761, 433856895597946272, 68964321078341276809, 12753724616472980432124, 2715405762438952565521785, 659549661987730244294458816, 181293528280954206831103494177

Offset: 1

Andrew Howroyd, Feb 25 2023

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..100

Row 2 of A360877.

Cf. A096121, A276356, A341500.

a(n)={sum(k=2, n, binomial(n,k)*k!) + sum(k=1, n, k*binomial(n,k)*binomial(k-1,k\2)*sum(i=0, n-k, binomial(n-k,i)*(k\2+i)!)*sum(i=0, n-k, binomial(n-k,i)*((k-1)\2+i)!))} \\ Andrew Howroyd, May 28 2025

a(n) = (Sum_{k=2..n} binomial(n,k)*k!) + (Sum_{k=1..n} k*binomial(n,k)*binomial(k-1, floor(k/2)) * (Sum_{i=0..n-k} binomial(n-k,i)*(floor(k/2)+i)!) * (Sum_{i=0..n-k} binomial(n-k,i)*(floor((k-1)/2)+i)!)). - Andrew Howroyd, May 28 2025

a(8) onwards from Andrew Howroyd, May 28 2025

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A269562 Array read by antidiagonals: T(n,m) is the number of Hamiltonian cycles in the rook graph K_n X K_m.

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A089039 Number of circular permutations of 2n letters that are free of jealousy.

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A360878 Number of (undirected) paths in the 2 X n rook graph.

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