A368257 - OEIS

A368257

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 an asymmetric tile.

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

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

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..36.

Peter Kagey, Illustration of T(2,2)=44

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

Adjacent sequences: A368254 A368255 A368256 * A368258 A368259 A368260

KEYWORD

nonn,tabl

AUTHOR

Peter Kagey, Dec 21 2023

STATUS

approved

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