|
%I #26 Apr 06 2020 06:29:11
%S 1,16,125,976,8512,79384,752061,7110272,67005561,630588698,5933085772,
%T 55827318685,525343024814,4943673540576,46521924780255,
%U 437788749723725,4119750109152730,38768318191017931,364823700357765771,3433121323699285343
%N Number of nonintersecting rook paths joining opposite corners of 5 X n board.
%D Netnews group rec.puzzles, Frequently Asked Questions (FAQ) file (Science Section).
%H Seiichi Manyama, <a href="/A007787/b007787.txt">Table of n, a(n) for n = 1..1000</a>
%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 D. G. Radcliffe, N. J. A. Sloane, C. Cole, J. Gillogly, & D. Dodson, <a href="/A007765/a007765.pdf">Emails, 1994</a>
%F Faase gives a 27-term linear recurrence on his web page:
%F a(1) = 1,
%F a(2) = 16,
%F a(3) = 125,
%F a(4) = 976,
%F a(5) = 8512,
%F a(6) = 79384,
%F a(7) = 752061,
%F a(8) = 7110272,
%F a(9) = 67005561,
%F a(10) = 630588698,
%F a(11) = 5933085772,
%F a(12) = 55827318685,
%F a(13) = 525343024814,
%F a(14) = 4943673540576,
%F a(15) = 46521924780255,
%F a(16) = 437788749723725,
%F a(17) = 4119750109152730,
%F a(18) = 38768318191017931,
%F a(19) = 364823700357765771,
%F a(20) = 3433121323699285343,
%F a(21) = 32306898830469680384,
%F a(22) = 304019468350280601960,
%F a(23) = 2860931888452842047170,
%F a(24) = 26922391858409506569346,
%F a(25) = 253349332040459400463497,
%F a(26) = 2384107785665647075602841,
%F a(27) = 22435306570786253414376286 and
%F a(n) = 30a(n-1) - 383a(n-2) + 2772a(n-3) - 12378a(n-4) + 33254a(n-5)
%F - 40395a(n-6) - 44448a(n-7) + 239776a(n-8) - 274256a(n-9) - 180404a(n-10)
%F + 678758a(n-11) - 301650a(n-12) - 542266a(n-13) + 492472a(n-14) + 184306a(n-15)
%F - 225284a(n-16) - 102314a(n-17) + 25534a(n-18) + 97396a(n-19) + 10392a(n-20)
%F - 40292a(n-21) - 13218a(n-22) + 5328a(n-23) + 5376a(n-24) + 1822a(n-25)
%F + 319a(n-26) + 24a(n-27).
%F Asymptotics: a(n) ~ 0.115762181699251 * 9.4103574958247159212^n [From _Vaclav Kotesovec_, Aug 31 2012]
%o (Python)
%o # Using graphillion
%o from graphillion import GraphSet
%o import graphillion.tutorial as tl
%o def A064298(n, k):
%o if n == 1 or k == 1: return 1
%o universe = tl.grid(n - 1, k - 1)
%o GraphSet.set_universe(universe)
%o start, goal = 1, k * n
%o paths = GraphSet.paths(start, goal)
%o return paths.len()
%o def A007787(n):
%o return A064298(n, 5)
%o print([A007787(n) for n in range(1, 20)]) # _Seiichi Manyama_, Apr 06 2020
%Y Row 5 of A064298.
%Y Cf. A007764, A007786.
%K nonn,walk
%O 1,2
%A Heiner Marxen
%E More terms from _Ralf Stephan_, Mar 29 2004
%E Added recurrence from Faase's web page. - _N. J. A. Sloane_, Feb 03 2009
| 