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%I #21 Jul 13 2014 02:29:21
%S 1,2,35,2688,508277,163715822,79059439095,53364540054860,
%T 47974697008198313,55410773910104281242,79957746695043660483467,
%U 140965507420235075126987480,298142048193613276717321211805,745056978435827991570581878537478
%N Number of distinct Tsuro tiles which are square and have n points per side.
%C Turning over is not allowed, but rotation of the tile is allowed.
%C In the original Tsuro game the tiles are square and have two points on each side, one third and two thirds of the way along the side and arcs connecting these eight points in various ways.
%C The shapes of the arcs aren't significant, only which two points they connect is.
%C Each point is connected to exactly one other point.
%C There are 35 tiles, agreeing with the entry a(4) = 35 here.
%C If we vary the shape of the tile and the number of points per side (pps), we get the following table.
%C ....pps:..0....1......2......3......4......5......6......7......8......9.....10
%C -------------------------------------------------------------------------------
%C circle....1....0......1......0......2......0......5......0.....18......0....105 (A007769)
%C monogon...1....0......1......0......3......0.....15......0....105......0....945 (A001147)
%C digon.....1....1......3.....11.....65....513...5363..68219 .................... (A132101)
%C triangle..1....0......7......0...3483......0.............0.............0
%C square....1....2.....35...2688.508277 ......................................... (this entry)
%C pentagon..1....0....193......0.............0.............0.............0
%C hexagon...1....5...1799
%C heptagon..1....0..19311......0.............0.............0.............0
%C octagon...1...18.254143
%C 9-gon.....1....0.............0.............0.............0.............0
%C 10-gon....1..105
%C The pps = 2 column is A132102. Blank entries all represent numbers greater than one million.
%C A monogon is distinct from a circle in that a monogon has not just one side, but also one vertex. Monogons and digons can't exist with straight sides, of course, at least not on a flat plane, but there's no rule that says these tiles have to have straight sides.
%C If we allow reflections the numbers are smaller (this would be appropriate for a game where the tiles were transparent and could be flipped over):
%C ....pps:..0....1......2......3......4......5......6......7......8......9.....10
%C -------------------------------------------------------------------------------
%C circle....1....0......1......0......2......0......5......0.....17......0.....79 (A054499)
%C monogon...1....0......1......0......3......0.....11......0.....65......0....513 (A132101)
%C digon.....1....1......3......8.....45....283...2847..34518.511209 ............. (A132103)
%C triangle..1....0......7......0...1907......0.............0.............0
%C square....1....2.....30...1447.257107 ......................................... (A132104)
%C pentagon..1....0....137......0.............0.............0.............0
%C hexagon...1....5...1065
%C heptagon..1....0..10307......0.............0.............0.............0
%C octagon...1...17.130040
%C 9-gon.....1....0.............0.............0.............0.............0
%C 10-gon....1...79
%C The pps = 2 column is A132105.
%H Calliope Games, <a href="http://www.calliopegames.com/read/45/tsuro">Tsuro</a>
%H Mike Garrity, <a href="http://blog.garritys.org/2012/01/path-tile-games.html">Path Tile Games</a>, Jan 07 2012.
%F From _Laurent Tournier_, Jul 09 2014: (Start)
%F a(m) = ((4m-1)!! + sum_{j=0..m} 2^j binomial(2m,2j) (2j-1)!! + 2 sum_{0<=2j<=m} 4^j binomial(m, 2j) (2j-1)!!)/4
%F More generally, if A(n,m) is the number of n-sided tiles with m points per side (with nm even),
%F A(n,m) = 1/n sum_{n=pq} phi(p)*alpha(p,mq), phi = Euler's totient function,
%F alpha(p,r) = sum_{0 <= 2j <= r} p^j binomial(r,2j) (2j-1)!! if p even,
%F = p^(r/2) (r-1)!! if p odd.
%F If B(n,m) is the number of n-sided tiles with m points per side (with nm even), allowing reflections,
%F B(n,m) = (A(n,m) + alpha(2,mn/2))/2 if m even,
%F = (A(n,m) + alpha(2,mn/2)/2 + alpha(2,mn/2-1)/2)/2 if m odd.
%F (End)
%p # A(n,m) gives the number of n-sided tiles with m points per side (cf. comments)
%p # B(n,m) enumerates these tiles, also allowing reflections
%p with(numtheory): a:=(p,r)->piecewise(p mod 2 = 1,p^(r/2)*doublefactorial(r-1), sum(p^j*binomial(r, 2*j)*doublefactorial(2*j - 1), j = 0 .. floor(r/2)));
%p A := (n,m)->piecewise(n*m mod 2=1,0,add(phi(p)*a(p,m*n/p),p in divisors(n))/n);
%p B := (n,m)->A(n,m)/2+piecewise(n*m mod 2=0,piecewise(m mod 2=0,a(2,m*n/2)*2, a(2,m*n/2)+a(2,m*n/2-1))/4,0);
%p A132100 := m -> A(4,m);[seq(A132100(m),m=1..15)]; # _Laurent Tournier_, Jul 09 2014
%Y Cf. A132101-A132105, A007769, A001147, A054499.
%K nonn
%O 0,2
%A _Keith F. Lynch_, Oct 31 2007
%E More terms from _Laurent Tournier_, Jul 09 2014
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