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Number of nonintersecting (or self-avoiding) rook paths joining opposite corners of 3 X n board.
(Formerly M3453)
7

%I M3453 #45 Sep 08 2022 08:44:34

%S 1,4,12,38,125,414,1369,4522,14934,49322,162899,538020,1776961,

%T 5868904,19383672,64019918,211443425,698350194,2306494009,7617832222,

%U 25159990674,83097804242,274453403399,906458014440

%N Number of nonintersecting (or self-avoiding) rook paths joining opposite corners of 3 X n board.

%D H. L. Abbott and D. Hanson, A lattice path problem, Ars Combin., 6 (1978), 163-178.

%D S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 331-339.

%D Netnews group rec.puzzles, Frequently Asked Questions (FAQ) file. (Science Section).

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

%H Vincenzo Librandi, <a href="/A006192/b006192.txt">Table of n, a(n) for n = 1..150</a>

%H H. L. Abbott and D. Hanson, <a href="/A006189/a006189.pdf">A lattice path problem</a>, Ars Combin., 6 (1978), 163-178. (Annotated scanned copy)

%H F. Faase, <a href="http://www.iwriteiam.nl/counting.html">Counting Hamiltonian cycles in product graphs</a>

%H F. Faase, <a href="http://www.iwriteiam.nl/Cresults.html">Results from the counting program</a>

%H Steven R. Finch, <a href="/FinchGammel.html">Self-Avoiding Walks of a Rook on a Chessboard</a> [From Steven Finch, Apr 20 2019]

%H Steven R. Finch, <a href="/FinchFlajolet.html">Self-Avoiding Walks of a Rook</a> [From Steven Finch, Apr 20 2019; mentioned in Finch's "Gammel" link above]

%H Steven R. Finch, <a href="/FinchMarxen.html">Table of Non-Overlapping Rook Paths</a> [From Steven Finch, Apr 20 2019; mentioned in Finch's "Gammel" link above]

%H D. G. Radcliffe, N. J. A. Sloane, C. Cole, J. Gillogly, & D. Dodson, <a href="/A007765/a007765.pdf">Emails, 1994</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,2,1).

%F a(n) = 4*a(n-1) - 3*a(n-2) + 2*a(n-3) + a(n-4) with a(0) = 0, a(1) = 1, a(2) = 4 and a(3) = 12. - _Henry Bottomley_, Sep 05 2001

%F G.f.: x*(1-x^2)/(1 - 4*x + 3*x^2 - 2*x^3 - x^4). - _Emeric Deutsch_, Dec 22 2004

%t LinearRecurrence[{4,-3,2,1},{1,4,12,38},40] (* _Harvey P. Dale_, Oct 05 2011 *)

%o (Magma) I:=[1,4,12,38]; [n le 4 select I[n] else 4*Self(n-1)-3*Self(n-2)+2*Self(n-3)+Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Oct 06 2011

%Y Cf. A064297, A064298, A007786, A007787, A007764.

%K nonn,walk,nice,easy

%O 1,2

%A _N. J. A. Sloane_