A222187 -id:A222187

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A222188

Table read by antidiagonals: number of toroidal m X n binary arrays, allowing rotation and/or reflection of the rows and/or the columns.

+10
12

2, 3, 3, 4, 7, 4, 6, 13, 13, 6, 8, 34, 36, 34, 8, 13, 78, 158, 158, 78, 13, 18, 237, 708, 1459, 708, 237, 18, 30, 687, 4236, 14676, 14676, 4236, 687, 30, 46, 2299, 26412, 184854, 340880, 184854, 26412, 2299, 46

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

G. C. Greubel, Table of n, a(n) for the first 100 antidiagonals, flattened

S. N. Ethier, Counting toroidal binary arrays, arXiv:1301.2352 [math.CO], 2013 and J. Int. Seq. 16 (2013) #13.4.7.

S. N. Ethier and Jiyeon Lee, Counting toroidal binary arrays, II, arXiv:1502.03792v1 [math.CO], Feb 12, 2015 and J. Int. Seq. 18 (2015) # 15.8.3.

S. N. Ethier and Jiyeon Lee, Parrondo games with two-dimensional spatial dependence, arXiv:1510.06947 [math.PR], 2015.

Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023. See p. 3.

EXAMPLE

Array begins:

2, 3, 4, 6, 8, 13, 18, 30, ...

3, 7, 13, 34, 78, 237, 687, 2299, ...

4, 13, 36, 158, 708, 4236, 26412, 180070, ...

6, 34, 158, 1459, 14676, 184854, 2445918, 33888844, ...

8, 78, 708, 14676, 340880, 8999762, 245619576, 6873769668, ...

...

MATHEMATICA

b1[m_, n_] := Sum[EulerPhi[c]*EulerPhi[d]*2^(m*n/LCM[c, d]), {c, Divisors[ m]}, {d, Divisors[n]}]/(4*m*n); b2a[m_, n_] := If[OddQ[m], 2^((m+1)*n/2) /(4*n), (2^(m*n/2) + 2^((m+2)*n/2))/(8*n)]; b2b[m_, n_] := DivisorSum[n, If[# >= 2, EulerPhi[#]*2^((m*n)/#), 0]&]/(4*n); b2c[m_, n_] := If[OddQ[ m], Sum[If [OddQ[n/GCD[j, n]], 2^((m+1)*GCD[j, n]/2) - 2^(m*GCD[j, n]), 0], {j, 1, n-1}]/(4*n), Sum[If[OddQ[n/GCD[j, n]], 2^(m*GCD[j, n]/2) + 2^((m+2)*GCD[j, n]/2) - 2^(m*GCD[j, n]+1), 0], {j, 1, n-1}]/(8*n)]; b2[m_, n_] := b2a[m, n] + b2b[m, n] + b2c[m, n]; b3[m_, n_] := b2[n, m]; b4oo[m_, n_] := 2^((m*n-3)/2); b4eo[m_, n_] := 3*2^(m*n/2 - 3); b4ee[m_, n_] := 7*2^(m*n/2-4); a[m_, n_] := Module[{b}, If [OddQ[m], If [OddQ[n], b = b4oo[m, n], b = b4eo[m, n]], If[OddQ[n], b = b4eo[m, n], b = b4ee[m, n]]]; b += b1[m, n] + b2[m, n] + b3[m, n]; Return[b]]; Table[a[m - n+1, n], {m, 1, 10}, {n, 1, m}] // Flatten (* Jean-François Alcover, Dec 05 2015, adapted from Michel Marcus's PARI script *)

PROG

(PARI)

odd(n) = n%2;

b1(m, n) = sumdiv(m, c, sumdiv(n, d, eulerphi(c)*eulerphi(d)*2^(m*n/lcm(c, d))))/(4*m*n);

b2a(m, n) = if (odd(m), 2^((m+1)*n/2)/(4*n), (2^(m*n/2)+2^((m+2)*n/2))/(8*n));

b2b(m, n) = sumdiv(n, d, if (d>=2, eulerphi(d)*2^((m*n)/d), 0))/(4*n);

b2c(m, n) = if (odd(m), sum(j=1, n-1, if (odd(n/gcd(j, n)), 2^((m+1)*gcd(j, n)/2)-2^(m*gcd(j, n))))/(4*n), sum(j=1, n-1, if (odd(n/gcd(j, n)), 2^(m*gcd(j, n)/2)+2^((m+2)*gcd(j, n)/2)-2^(m*gcd(j, n)+1)))/(8*n));

b2(m, n) = b2a(m, n) + b2b(m, n) + b2c(m, n);

b3(m, n) = b2(n, m);

b4oo(m, n) = 2^((m*n - 3)/2);

b4eo(m, n) = 3*2^(m*n/2 - 3);

b4ee(m, n) = 7*2^(m*n/2 - 4);

a(m, n) = {if (odd(m), if (odd(n), b = b4oo(m, n), b = b4eo(m, n)), if (odd(n), b = b4eo(m, n), b = b4ee(m, n))); b += b1(m, n) + b2(m, n) + b3(m, n); return (b); }

\\ Michel Marcus, Feb 13 2013

CROSSREFS

Rows give A000029, A222187, A222189, A222190, A222191.

Main diagonal is A209251.

Cf. A184271.

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Feb 12 2013

STATUS

approved

A368253

Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal and vertical reflections by two tiles that are fixed under these reflections.

+10
3

2, 3, 3, 6, 7, 4, 10, 24, 13, 6, 20, 76, 74, 34, 8, 36, 288, 430, 378, 78, 13, 72, 1072, 3100, 4756, 1884, 237, 18, 136, 4224, 23052, 70536, 53764, 11912, 687, 30, 272, 16576, 179736, 1083664, 1689608, 709316, 77022, 2299, 46

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..45.

Peter Kagey, Illustration of T(2,3)=24

Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023.

EXAMPLE

Table begins:

n\k | 1 2 3 4 5 6

----+----------------------------------------

1 | 2 3 6 10 20 36

2 | 3 7 24 76 288 1072

3 | 4 13 74 430 3100 23052

4 | 6 34 378 4756 70536 1083664

5 | 8 78 1884 53764 1689608 53762472

6 | 13 237 11912 709316 44900448 2865540112

MATHEMATICA

A368253[n_, m_] := 1/(4n)*(DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/d)]] + n*If[EvenQ[n], 1/2 (2^((n*m + 2 m)/2) + 2^(n*m/2)), 2^((n*m + m)/2)] + If[EvenQ[m], DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/LCM[d, 2])]], DivisorSum[n, Function[d, EulerPhi[d]*2^((n*m - n)/LCM[d, 2])*2^(n/d)]]] + n*Which[EvenQ[m], 2^(n*m/2), OddQ[m] && EvenQ[n], (3/2*2^(n*m/2)), OddQ[m] && OddQ[n], 2^((n*m + 1)/2)])

CROSSREFS

Cf. A222188, A225910.

Cf. A005418 (n=1), A225826 (n=2), A000029 (k=1), A222187 (k=2).