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A132100 Number of distinct Tsuro tiles which are square and have n points per side. 6
1, 2, 35, 2688, 508277, 163715822, 79059439095, 53364540054860, 47974697008198313, 55410773910104281242, 79957746695043660483467, 140965507420235075126987480, 298142048193613276717321211805, 745056978435827991570581878537478 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Turning over is not allowed, but rotation of the tile is allowed.

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.

The shapes of the arcs aren't significant, only which two points they connect is.

Each point is connected to exactly one other point.

There are 35 tiles, agreeing with the entry a(4) = 35 here.

If we vary the shape of the tile and the number of points per side (pps), we get the following table.

....pps:..0....1......2......3......4......5......6......7......8......9.....10

-------------------------------------------------------------------------------

circle....1....0......1......0......2......0......5......0.....18......0....105 (A007769)

monogon...1....0......1......0......3......0.....15......0....105......0....945 (A001147)

digon.....1....1......3.....11.....65....513...5363..68219 .................... (A132101)

triangle..1....0......7......0...3483......0.............0.............0

square....1....2.....35...2688.508277 ......................................... (this entry)

pentagon..1....0....193......0.............0.............0.............0

hexagon...1....5...1799

heptagon..1....0..19311......0.............0.............0.............0

octagon...1...18.254143

9-gon.....1....0.............0.............0.............0.............0

10-gon....1..105

The pps = 2 column is A132102. Blank entries all represent numbers greater than one million.

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.

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):

....pps:..0....1......2......3......4......5......6......7......8......9.....10

-------------------------------------------------------------------------------

circle....1....0......1......0......2......0......5......0.....17......0.....79 (A054499)

monogon...1....0......1......0......3......0.....11......0.....65......0....513 (A132101)

digon.....1....1......3......8.....45....283...2847..34518.511209 ............. (A132103)

triangle..1....0......7......0...1907......0.............0.............0

square....1....2.....30...1447.257107 ......................................... (A132104)

pentagon..1....0....137......0.............0.............0.............0

hexagon...1....5...1065

heptagon..1....0..10307......0.............0.............0.............0

octagon...1...17.130040

9-gon.....1....0.............0.............0.............0.............0

10-gon....1...79

The pps = 2 column is A132105.

LINKS

Table of n, a(n) for n=0..13.

Calliope Games, Tsuro

Mike Garrity, Path Tile Games, Jan 07 2012.

FORMULA

From Laurent Tournier, Jul 09 2014: (Start)

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

More generally, if A(n,m) is the number of n-sided tiles with m points per side (with nm even),

A(n,m) = 1/n sum_{n=pq} phi(p)*alpha(p,mq), phi = Euler's totient function,

alpha(p,r) = sum_{0 <= 2j <= r} p^j binomial(r,2j) (2j-1)!! if p even,

           = p^(r/2) (r-1)!! if p odd.

If B(n,m) is the number of n-sided tiles with m points per side (with nm even), allowing reflections,

B(n,m) = (A(n,m) + alpha(2,mn/2))/2 if m even,

       = (A(n,m) + alpha(2,mn/2)/2 + alpha(2,mn/2-1)/2)/2 if m odd.

(End)

MAPLE

# A(n, m) gives the number of n-sided tiles with m points per side (cf. comments)

# B(n, m) enumerates these tiles, also allowing reflections

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)));

A := (n, m)->piecewise(n*m mod 2=1, 0, add(phi(p)*a(p, m*n/p), p in divisors(n))/n);

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);

A132100 := m -> A(4, m); [seq(A132100(m), m=1..15)]; # Laurent Tournier, Jul 09 2014

CROSSREFS

Cf. A132101-A132105, A007769, A001147, A054499.

Sequence in context: A191807 A245052 A110697 * A182070 A133013 A032369

Adjacent sequences:  A132097 A132098 A132099 * A132101 A132102 A132103

KEYWORD

nonn

AUTHOR

Keith F. Lynch, Oct 31 2007

EXTENSIONS

More terms from Laurent Tournier, Jul 09 2014

STATUS

approved

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Last modified October 22 11:28 EDT 2014. Contains 248397 sequences.