A132102 -id:A132102 - cp's OEIS frontend

A132101 a(n) = (A001147(n) + A047974(n))/2.

1, 1, 3, 11, 65, 513, 5363, 68219, 1016481, 17243105, 327431363, 6874989963, 158118876449, 3952936627361, 106729080101235, 3095142009014843, 95949394016339393, 3166329948046914369, 110821547820208233731, 4100397266856761733515

Offset: 0

Keith F. Lynch, Oct 31 2007

Also, number of distinct Tsuro tiles which are digonal in shape and have n points per side. Turning over is not allowed. See A132100 for definition and comments.
See the Burns et al. papers for another interpretation.
From Ross Drewe, Mar 16 2008: (Start)
This is also the number of arrangements of n pairs which are equivalent under the joint operation of sequence reversal and permutations of labels. Assume that the elements of n distinct pairs are labeled to show the pair of origin, e.g., [1 1], [2 2]. The number of distinguishable ways of arranging these elements falls as the conditions are made more general:
a(n) = A000680: element order is significant and the labels are distinguishable;
b(n) = A001147: element order is significant but labels are not distinguishable, i.e., all label permutations of a given sequence are equivalent;
c(n) = A132101: element order is weakened (reversal allowed) and all label permutations are equivalent;
d(n) = A047974: reversal allowed, all label permutations are equivalent and equivalence class maps to itself under joint operation.
Those classes that do not map to themselves form reciprocal pairs of classes under the joint operation and their number is r(n). Then c = b - r/2 = b - (b - d)/2 = (b+d)/2. A formula for r(n) is not available, but formulas are available for b(n) = A001147 and d(n) = A047974, allowing an explicit formula for this sequence.
c(n) is useful in extracting structure information without regard to pair ordering (see example). c(n) terms also appear in formulas related to binary operators, e.g., the number of binary operators in a k-valued logic that are invertible in 1 operation.
a(n) = (b(n) + c(n))/2, where b(n) = (2n)!/(2^n * n!) = A001147(n), c(n) = Sum_{k=0..floor(n/2)} n!/((n-2*k)! * k!) = A047974(n).
For 3 pairs, the arrangement A = [112323] is the same as B = [212133] under the permutation of the labels [123] -> [312] plus reversal of the elements, or vice versa. The unique structure common to A and B is {1 intact pair + 2 interleaved pairs}, where the order is not significant (contrast A001147). (End)

			a(2)=3 counts the arrangements [1122], [1212] and [1221]. - _R. J. Mathar_, Oct 18 2019

G. C. Greubel, Table of n, a(n) for n = 0..400
Jonathan Burns, Assembly Graph Words - Single Transverse Component (Counts).
Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche, and Masahico Saito, Four-Regular Graphs with Rigid Vertices Associated to DNA Recombination, Discrete Applied Mathematics, Volume 161, Issues 10-11, July 2013, Pages 1378-1394.
R. J. Mathar, Feynman diagrams of the QED vacuum polarization, vixra:1901.0148 (2019), Section IV.

Cf. A000680, A001147, A047974, A007769, A054499.

Cf. A132100, A132101, A132102, A132103, A132104, A132105.

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( (Exp(x+x^2) + 1/Sqrt(1-2*x))/2 ))); // G. C. Greubel, Jul 12 2024

A132101 := proc(n)
    (A001147(n)+A047974(n))/2 ;
end proc:
seq(A132101(n),n=0..30) ; # R. J. Mathar, Dec 20 2020

Table[((2n-1)!!+I^(-n)*HermiteH[n,I/2])/2,{n,0,30}] (* Jonathan Burns, Apr 05 2016 *)

[(factorial(n)*binomial(2*n,n) + (-2*i)^n*hermite(n,i/2))/2^(n+1) for n in range(31)] # G. C. Greubel, Jul 12 2024

D-finite with recurrence -(n-3)*a(n) +2*(n^2-3*n+1)*a(n-1) -(n-1)*a(n-2) -2*(2*n-5)*(n-1)*(n-2)*a(n-3) = 0. - R. J. Mathar, Dec 20 2020

E.g.f.: (1/2)*( exp(x+x^2) + 1/sqrt(1-2*x) ). - G. C. Greubel, Jul 12 2024

Entry revised by N. J. A. Sloane, Nov 04 2011

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

Original entry on oeis.org

1, 2, 35, 2688, 508277, 163715822, 79059439095, 53364540054860, 47974697008198313, 55410773910104281242, 79957746695043660483467, 140965507420235075126987480, 298142048193613276717321211805, 745056978435827991570581878537478

Offset: 0

Keith F. Lynch, Oct 31 2007

nonn

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.

Calliope Games, Tsuro
Mike Garrity, Path Tile Games, Jan 07 2012.

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

# 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

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)

More terms from Laurent Tournier, Jul 09 2014

A289191 Number of polygonal tiles with n sides with two exits per side and n edges connecting pairs of exits, with no edges between exits on the same side and non-isomorphic under rotational symmetry.

Original entry on oeis.org

0, 2, 4, 22, 112, 1060, 11292, 149448, 2257288, 38720728, 740754220, 15648468804, 361711410384, 9081485302372, 246106843197984, 7160143986526240, 222595582448849152, 7364186944683168828, 258327454310582805036, 9577476294162996275928, 374205233351106756670120

Offset: 1

Marko Riedel, Jun 27 2017

nonn

The case n=2 is a degenerate polygon (two sides connecting two vertices). The two possibilities are when the edges cross and do not cross. Polygons start at n=3 with a triangle.

The sequence A132102 enumerates the case that edges are allowed between exits on the same side. This sequence can be enumerated in a similar manner using inclusion-exclusion on the number of sides that have their two exits connected. - Andrew Howroyd, Jan 26 2020

Andrew Howroyd, Table of n, a(n) for n = 1..200
Marko Riedel, Hexagonal tiles.
Marko Riedel, Maple code to compute number of tiles by ordinary enumeration and by Power Group Enumeration.
Marko Riedel, Maple code for number of tiles using Burnside lemma.

See A053871 for tiles with no rotational symmetries being taken into account, A289269 for tiles with rotational and reflectional symmetries being taken into account, A289343 for the same statistic evaluated when n is prime.

Cf. A132102.

a(n) = {sumdiv(n, d, my(m=n/d); eulerphi(d)*sum(i=0, m, (-1)^i * binomial(m, i) * sum(j=0, m-i, (d%2==0 || m-i-j==0) * binomial(2*(m-i), 2*j) * d^j * (2*j)! / (j!*2^j) )))/n} \\ Andrew Howroyd, Jan 26 2020

Terms a(14) and beyond from Andrew Howroyd, Jan 26 2020

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A132101 a(n) = (A001147(n) + A047974(n))/2.

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

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A289191 Number of polygonal tiles with n sides with two exits per side and n edges connecting pairs of exits, with no edges between exits on the same side and non-isomorphic under rotational symmetry.

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