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%I #72 Aug 06 2023 14:16:55
%S 1,8,40,552,8648,458696,27070560,6046626568,1490832682992,
%T 1460089659025264,1573342970540617696,6905329711608694708440,
%U 33304011435341069362631160,663618176813467308855850585056,14527222735920532980525200234503048
%N 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).
%C Number of directed Hamiltonian paths in (n X n)-grid graph. - _Max Alekseyev_, May 03 2009
%H Andrew Howroyd, <a href="/A096969/b096969.txt">Table of n, a(n) for n = 1..17</a>
%H Nicolas Bělohoubek, <a href="/A096969/a096969.txt">All possible paths in 4th term (552) presented by A..D 1..4 coordination system.</a>
%H Nicolas Bělohoubek, <a href="/A096969/a096969_1.txt">All possible paths in 5th term (8648) presented by A..E 1..5 coordination system.</a>
%H Nicolas Bělohoubek, <a href="/A096969/a096969.png">All possible paths in 5th term (8648) in image, blue to red.</a>
%H Stéphane Duguay and Steven Pigeon, <a href="http://stevenpigeon.com/Publications/publications/Duguay--Comparison-of-Pixel-Correlation-Induced-by-SFCs--2019--DRAFT.pdf">Comparison of Pixel Correlation Induced by Space-Filling Curves on 2D Image Data</a>, The 10th IEEE International Conference on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications (Metz, France, 2019) Vol. 1, 294-297.
%H Mary Grace Hanson and David A. Nash, <a href="https://arxiv.org/abs/1706.09389">Minimal and maximal Numbrix puzzles</a>, arXiv:1706.09389 [math.CO], 2017.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HamiltonianPath.html">Hamiltonian Path</a>
%H <a href="/index/Gra#graphs">Index entries for sequences related to graphs, Hamiltonian</a>
%F 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
%e One of the 8648 numberings of a 5 X 5 grid is
%e .
%e 3---2---1 20--21
%e | | |
%e 4 17--18--19 22
%e | | |
%e 5 16--15--14 23
%e | | |
%e 6 9--10 13 24
%e | | | | |
%e 7---8 11--12 25
%Y Cf. A120443, A096970, A143246, A137891, A158651.
%K nonn,walk
%O 1,2
%A _John W. Layman_, Jul 16 2004, at the suggestion of _Leroy Quet_, Jul 05 2004
%E a(7) from _Giovanni Resta_, May 12 2006
%E a(8)-a(15) added by _Andrew Howroyd_, Dec 20 2015
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