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A368257
Table read by antidiagonals downward: T(n,k) is the number of tilings of the n X k cylinder up to horizontal and vertical reflections by an asymmetric tile.
4
1, 6, 4, 16, 44, 6, 72, 544, 366, 23, 256, 8384, 21856, 4244, 52, 1056, 131584, 1399512, 1050128, 52740, 194, 4096, 2100224, 89478656, 268472384, 53687104, 701124, 586, 16512, 33562624, 5726711136, 68719870208, 54975896016, 2863399264, 9591666, 2131
OFFSET
1,2
LINKS
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
---+-------------------------------------------------------
1 | 1 6 16 72 256
2 | 4 44 544 8384 131584
3 | 6 366 21856 1399512 89478656
4 | 23 4244 1050128 268472384 68719870208
5 | 52 52740 53687104 54975896016 56294995342336
6 | 194 701124 2863399264 11728132423744 48038396383286784
MATHEMATICA
A368257[n_, m_] := 1/(4n)*(DivisorSum[n, EulerPhi[#]*4^(n*m/#) &] + n (2^(n*m - 1))*Boole[EvenQ[n]] + If[EvenQ[m], DivisorSum[n, EulerPhi[#]*4^(n*m/LCM[#, 2]) &], DivisorSum[n, EulerPhi[#]*4^(n*m/#) &, EvenQ]] + n*2^(n*m)*Which[EvenQ[m], 1, EvenQ[n], 1/2, True, 0])
CROSSREFS
Sequence in context: A198459 A083581 A171089 * A180495 A213761 A160248
KEYWORD
nonn,tabl
AUTHOR
Peter Kagey, Dec 21 2023
STATUS
approved