A137891 -id:A137891 - cp's OEIS frontend

A270047 Decimal expansion of a constant related to the asymptotics of A137891.

4, 2, 1, 2, 2, 7, 7, 4, 2, 1, 1, 6, 8, 1, 5, 6, 0, 8, 1, 1, 6, 6, 2, 9, 2, 5, 5, 0, 1, 0, 5, 9, 5, 6, 6, 7, 9, 4, 9, 3, 7, 1, 7, 3, 8, 5, 5, 3, 7, 6, 6, 1, 0, 0, 2, 5, 8, 6, 8, 0, 4, 8, 4, 8, 9, 8, 4, 0, 4, 6, 8, 8, 2, 3, 3, 7, 7, 0, 1, 4, 3, 1, 3, 9, 7, 2, 4, 8, 7, 6, 5, 0, 3, 4, 0, 8, 8, 2, 4, 9, 5, 0, 1, 1, 2, 7

Offset: 2

Vaclav Kotesovec, Mar 09 2016

			42.12277421168156081166292550105956679493717385537661002586804848984...

Eric Weisstein's World of Mathematics, Graph Join
Eric Weisstein's World of Mathematics, Hamiltonian Path

Cf. A137891.

RealDigits[2*(BesselI[0, 4] + BesselI[1, 4]), 10, 120][[1]] (* Vaclav Kotesovec, Mar 16 2024 *)

Equals limit n->infinity A137891(n)/(n!)^2.

Equals 2*(BesselI(0,4) + BesselI(1,4)). - Vaclav Kotesovec, Mar 16 2024

A158651 Number of directed Hamiltonian paths on the n X n king graph.

Original entry on oeis.org

1, 24, 784, 343184, 729237344, 13089822163800, 1659110130720710584, 1635069460917798701270872, 12308784500036123518164726610224, 721833220650131890343295654587745095696, 330596986686626406483380599328509951896788808144

Offset: 1

Eric W. Weisstein, Mar 23 2009

Number of open directed king's tours on the n X n board.

Ville H. Pettersson, Enumerating Hamiltonian Cycles, The Electronic Journal of Combinatorics, Volume 21, Issue 4, 2014.
Eric Weisstein's World of Mathematics, Hamiltonian Path
Eric Weisstein's World of Mathematics, King Graph
Index entries for sequences related to graphs, Hamiltonian

Cf. A003763, A096969, A137891, A140521.

a(5) from Max Alekseyev, May 03 2009

a(6)-a(11) from Andrew Howroyd, Nov 15 2015

A096969 Number of ways to number the cells of an n X n square grid with 1,2,3,...,n^2 so that successive integers are in adjacent cells (horizontally or vertically).

Original entry on oeis.org

1, 8, 40, 552, 8648, 458696, 27070560, 6046626568, 1490832682992, 1460089659025264, 1573342970540617696, 6905329711608694708440, 33304011435341069362631160, 663618176813467308855850585056, 14527222735920532980525200234503048

Offset: 1

John W. Layman, Jul 16 2004, at the suggestion of Leroy Quet, Jul 05 2004

Number of directed Hamiltonian paths in (n X n)-grid graph. - Max Alekseyev, May 03 2009

			One of the 8648 numberings of a 5 X 5 grid is
.
  3---2---1  20--21
  |           |   |
  4  17--18--19  22
  |   |           |
  5  16--15--14  23
  |           |   |
  6   9--10  13  24
  |   |   |   |   |
  7---8  11--12  25

Andrew Howroyd, Table of n, a(n) for n = 1..17
Nicolas Bělohoubek, All possible paths in 4th term (552) presented by A..D 1..4 coordination system.
Nicolas Bělohoubek, All possible paths in 5th term (8648) presented by A..E 1..5 coordination system.
Nicolas Bělohoubek, All possible paths in 5th term (8648) in image, blue to red.
Stéphane Duguay and Steven Pigeon, Comparison of Pixel Correlation Induced by Space-Filling Curves on 2D Image Data, The 10th IEEE International Conference on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications (Metz, France, 2019) Vol. 1, 294-297.
Mary Grace Hanson and David A. Nash, Minimal and maximal Numbrix puzzles, arXiv:1706.09389 [math.CO], 2017.
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Hamiltonian Path
Index entries for sequences related to graphs, Hamiltonian

Cf. A120443, A096970, A143246, A137891, A158651.

Conjecture: Limit_{n->oo} log_(n+1)!(a(n+1)) - log_n!(a(n)) = c, where 0.09 < c < 0.11. - Nicolas Bělohoubek, Jun 12 2022

a(7) from Giovanni Resta, May 12 2006

a(8)-a(15) added by Andrew Howroyd, Dec 20 2015

A234628 Number of undirected cycles in the graph join C_n + C_n of cycle graphs.

Original entry on oeis.org

197, 2766, 60142, 1921115, 84779781, 4939361970, 367260392738, 33943163219867, 3817771391967469, 513555324752795822, 81424493891629268382, 15029177059618712556883, 3195217968081609415815677, 775224832686819672474135234, 212905295555773231486108386402

Offset: 3

Eric W. Weisstein, Dec 28 2013

nonn

Andrew Howroyd, Table of n, a(n) for n = 3..100
Eric Weisstein's World of Mathematics, Graph Cycle.
Eric Weisstein's World of Mathematics, Graph Join.

Cf. A137891.

B(n)=polcoef(1/(1 - x*y*(2/(1 - x) - 1)/(1 - x)) + O(x*x^n), n)
a(n)={my(v=Vecrev(B(n))); (sum(k=1, n, (n*v[1+k]/k)^2*k!*(k-1)!) + 4 - n^2)/2} \\ Andrew Howroyd, Jan 10 2025

a(n) = 2 + n^2*(-1 + Sum_{k=1..n} ((k-1)!*B(n,k))^2/k)/2, where B(n,k) = [x^n][y^k] 1/(1 - x*y*(2/(1 - x) - 1)/(1 - x)). - Andrew Howroyd, Jan 10 2025

Name corrected by Eric W. Weisstein, Mar 09 2016
a(11)-a(15) from Max Alekseyev, Dec 30 2024
a(2) removed and a(16) onwards added by Andrew Howroyd, Jan 10 2025

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A270047 Decimal expansion of a constant related to the asymptotics of A137891.

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A158651 Number of directed Hamiltonian paths on the n X n king graph.

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A096969 Number of ways to number the cells of an n X n square grid with 1,2,3,...,n^2 so that successive integers are in adjacent cells (horizontally or vertically).

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A234628 Number of undirected cycles in the graph join C_n + C_n of cycle graphs.

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