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A132100 Number of distinct Tsuro tiles which are square and have n points per side.

%I A132100 #21 Jul 13 2014 02:29:21
%S A132100 1,2,35,2688,508277,163715822,79059439095,53364540054860,
%T A132100 47974697008198313,55410773910104281242,79957746695043660483467,
%U A132100 140965507420235075126987480,298142048193613276717321211805,745056978435827991570581878537478
%N A132100 Number of distinct Tsuro tiles which are square and have n points per side.
%C A132100 Turning over is not allowed, but rotation of the tile is allowed.
%C A132100 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 A132100 The shapes of the arcs aren't significant, only which two points they connect is.
%C A132100 Each point is connected to exactly one other point.
%C A132100 There are 35 tiles, agreeing with the entry a(4) = 35 here.
%C A132100 If we vary the shape of the tile and the number of points per side (pps), we get the following table.
%C A132100 ....pps:..0....1......2......3......4......5......6......7......8......9.....10
%C A132100 -------------------------------------------------------------------------------
%C A132100 circle....1....0......1......0......2......0......5......0.....18......0....105 (A007769)
%C A132100 monogon...1....0......1......0......3......0.....15......0....105......0....945 (A001147)
%C A132100 digon.....1....1......3.....11.....65....513...5363..68219 .................... (A132101)
%C A132100 triangle..1....0......7......0...3483......0.............0.............0
%C A132100 square....1....2.....35...2688.508277 ......................................... (this entry)
%C A132100 pentagon..1....0....193......0.............0.............0.............0
%C A132100 hexagon...1....5...1799
%C A132100 heptagon..1....0..19311......0.............0.............0.............0
%C A132100 octagon...1...18.254143
%C A132100 9-gon.....1....0.............0.............0.............0.............0
%C A132100 10-gon....1..105
%C A132100 The pps = 2 column is A132102. Blank entries all represent numbers greater than one million.
%C A132100 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 A132100 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 A132100 ....pps:..0....1......2......3......4......5......6......7......8......9.....10
%C A132100 -------------------------------------------------------------------------------
%C A132100 circle....1....0......1......0......2......0......5......0.....17......0.....79 (A054499)
%C A132100 monogon...1....0......1......0......3......0.....11......0.....65......0....513 (A132101)
%C A132100 digon.....1....1......3......8.....45....283...2847..34518.511209 ............. (A132103)
%C A132100 triangle..1....0......7......0...1907......0.............0.............0
%C A132100 square....1....2.....30...1447.257107 ......................................... (A132104)
%C A132100 pentagon..1....0....137......0.............0.............0.............0
%C A132100 hexagon...1....5...1065
%C A132100 heptagon..1....0..10307......0.............0.............0.............0
%C A132100 octagon...1...17.130040
%C A132100 9-gon.....1....0.............0.............0.............0.............0
%C A132100 10-gon....1...79
%C A132100 The pps = 2 column is A132105.
%H A132100 Calliope Games, <a href="http://www.calliopegames.com/read/45/tsuro">Tsuro</a>
%H A132100 Mike Garrity, <a href="http://blog.garritys.org/2012/01/path-tile-games.html">Path Tile Games</a>, Jan 07 2012.
%F A132100 From _Laurent Tournier_, Jul 09 2014: (Start)
%F A132100 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 A132100 More generally, if A(n,m) is the number of n-sided tiles with m points per side (with nm even),
%F A132100 A(n,m) = 1/n sum_{n=pq} phi(p)*alpha(p,mq), phi = Euler's totient function,
%F A132100 alpha(p,r) = sum_{0 <= 2j <= r} p^j binomial(r,2j) (2j-1)!! if p even,
%F A132100            = p^(r/2) (r-1)!! if p odd.
%F A132100 If B(n,m) is the number of n-sided tiles with m points per side (with nm even), allowing reflections,
%F A132100 B(n,m) = (A(n,m) + alpha(2,mn/2))/2 if m even,
%F A132100        = (A(n,m) + alpha(2,mn/2)/2 + alpha(2,mn/2-1)/2)/2 if m odd.
%F A132100 (End)
%p A132100 # A(n,m) gives the number of n-sided tiles with m points per side (cf. comments)
%p A132100 # B(n,m) enumerates these tiles, also allowing reflections
%p A132100 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 A132100 A := (n,m)->piecewise(n*m mod 2=1,0,add(phi(p)*a(p,m*n/p),p in divisors(n))/n);
%p A132100 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 A132100 := m -> A(4,m);[seq(A132100(m),m=1..15)]; # _Laurent Tournier_, Jul 09 2014
%Y A132100 Cf. A132101-A132105, A007769, A001147, A054499.
%K A132100 nonn
%O A132100 0,2
%A A132100 _Keith F. Lynch_, Oct 31 2007
%E A132100 More terms from _Laurent Tournier_, Jul 09 2014