A112841 -id:A112841 - cp's OEIS frontend

A112844 Small-number statistic from the enumeration of domino tilings of a 9-pillow of order n.

1, 2, 5, 13, 34, 89, 89, 193, 185, 410, 482, 1444, 2018, 6362, 8461, 19885, 22861, 51125, 59792, 146749, 195749, 529114, 730465, 1907545, 2350177, 5638489, 6692337, 16167545, 20091490, 51762100, 67753160, 178151440, 229118152

Offset: 0

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005

A 9-pillow is a generalized Aztec pillow. The 9-pillow of order n is a rotationally-symmetric region. It has a 2 X 2n central band of squares and then steps up from this band with steps of 9 horizontal squares to every 1 vertical square and steps down with steps of 1 horizontal square to every 1 vertical square.

Plotting A112844(n+2)/A112844(n) gives an intriguing damped sine curve.

			The number of domino tilings of the 9-pillow of order 8 is 9065=7^2*185. A112844(n)=185.

C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis. University of Washington, Seattle, USA.

A112842 breaks down as A112843^2 times A112844, where A112844 is not necessarily squarefree.

3-pillows: A112833-A112835; 5-pillows: A112836-A112838; 7-pillows: A112839-A112841.

A112833 Number of domino tilings of a 3-pillow of order n.

Original entry on oeis.org

1, 2, 5, 20, 117, 1024, 13357, 259920, 7539421, 326177280, 21040987113, 2024032315968, 290333133984905, 62102074862600192, 19808204598680574457, 9421371079480456587520, 6682097668647718038428569, 7067102111711681259234263040, 11145503882824383823706372042925

Offset: 0

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005

nonn

A 3-pillow is also called an Aztec pillow. The 3-pillow of order n is a rotationally-symmetric region. It has a 2 X 2n central band of squares and then steps up from this band with steps of 3 horizontal squares to every 1 vertical square and steps down with steps of 1 horizontal square to every 1 vertical square.

a(n)^(1/n^2) tends to 1.2211384384439007690866503099... - Vaclav Kotesovec, May 19 2020

			The number of domino tilings of the 3-pillow of order 4 is 117=3^2*13.

Alois P. Heinz, Table of n, a(n) for n = 0..100
C. Hanusa, A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows, PhD Thesis, 2005, University of Washington, Seattle, USA.

This sequence breaks down as A112834^2 times A112835, where A112835 is not necessarily squarefree.
5-pillows: A112836-A112838; 7-pillows: A112839-A112841; 9-pillows: A112842-A112844.
Related to A071101 and A071100.

with(LinearAlgebra):
b:= proc(x, y, k) option remember;
      `if`(y>x or y Matrix(n, (i, j)-> b(i-1, i-1, j-1)):
R:= n-> Matrix(n, (i, j)-> `if`(i+j=n+1, 1, 0)):
a:= n-> Determinant(P(n)+R(n).(P(n)^(-1)).R(n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 26 2013

b[x_, y_, k_] := b[x, y, k] = If[y>x || yJean-François Alcover, Nov 08 2015, after Alois P. Heinz *)

A112835 Small-number statistic from the enumeration of domino tilings of a 3-pillow of order n.

Original entry on oeis.org

1, 2, 5, 5, 13, 16, 37, 45, 109, 130, 313, 377, 905, 1088, 2617, 3145, 7561, 9090, 21853, 26269, 63157, 75920, 182525, 219413, 527509, 634114, 1524529, 1832625, 4405969, 5296384, 12733489, 15306833, 36800465, 44237570, 106355317

Offset: 0

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005

A 3-pillow is also called an Aztec pillow. The 3-pillow of order n is a rotationally-symmetric region. It has a 2 X 2n central band of squares and then steps up from this band with steps of 3 horizontal squares to every 1 vertical square and steps down with steps of 1 horizontal square to every 1 vertical square.

Plotting A112835(n+2)/A112835(n) gives an intriguing damped sine curve.

			1 + 2*x + 5*x^2 + 5*x^3 + 13*x^4 + 16*x^5 + 37*x^6 + 45*x^7 + 109*x^8 + ...
The number of domino tilings of the 3-pillow of order 4 is 117=3^2*13. A112835(4)=13.

C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis. University of Washington, Seattle, USA.

A D Mednykh, I A Mednykh, The number of spanning trees in circulant graphs, its arithmetic properties and asymptotic, arXiv preprint arXiv:1711.00175, 2017. See Section 4.

A112833 breaks down as A112834^2 times A112835, where A112835 is not necessarily squarefree.
5-pillows: A112836-A112838; 7-pillows: A112839-A112841; 9-pillows: A112842-A112844.
Cf. A071100, A071101.

{a(n) = local(m = abs(n+3)); polcoeff( (x + x^2 - x^3 + x^5 - x^6 - x^7) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8)  + x * O(x^m), m)} /* Michael Somos, Dec 15 2011 */

a(2*n + 2) = A071100(n). a(2*n + 3) = A071101(n).
G.f.: (1 + x - x^2 + x^4 - x^5 - x^6) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8) = (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8). - Michael Somos, Dec 15 2011
a(-n) = a(-6 + n). a(-1) = a(-2) = 1, a(-3) = 0. a(n) = 2*a(n-2) + 2*a(n-4) + 2*a(n-6) - a(n-8). - Michael Somos, Dec 15 2011

A112839 Number of domino tilings of a 7-pillow of order n.

Original entry on oeis.org

1, 2, 5, 13, 34, 136, 666, 3577, 23353, 200704, 2062593, 24878084, 373006265, 6917185552, 153624835953, 4155902941554, 138450383756352, 5602635336941568, 274540864716936000, 16486029239132118530, 1209110712606533552257

Offset: 0

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005

nonn

A 7-pillow is a generalized Aztec pillow. The 7-pillow of order n is a rotationally-symmetric region. It has a 2 X 2n central band of squares and then steps up from this band with steps of 7 horizontal squares to every 1 vertical square and steps down with steps of 1 horizontal square to every 1 vertical square.

			The number of domino tilings of the 7-pillow of order 8 is 23353=11^2*193.

C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis. University of Washington, Seattle, USA.

A112839 breaks down as A112840^2 times A112841, where A112841 is not necessarily squarefree.

3-pillows: A112833-A112835; 5-pillows: A112836-A112838; 9-pillows: A112842-A112844.

A112842 Number of domino tilings of a 9-pillow of order n.

Original entry on oeis.org

1, 2, 5, 13, 34, 89, 356, 1737, 9065, 49610, 325832, 2795584, 28098632, 310726442, 3877921669, 58896208285, 1083370353616, 22901813128125, 548749450880000, 15471093192996501, 522297110942557556, 20691062026775504896

Offset: 0

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005

nonn

A 9-pillow is a generalized Aztec pillow. The 9-pillow of order n is a rotationally-symmetric region. It has a 2 X 2n central band of squares and then steps up from this band with steps of 9 horizontal squares to every 1 vertical square and steps down with steps of 1 horizontal square to every 1 vertical square.

			The number of domino tilings of the 9-pillow of order 8 is 9065=7^2*185.

C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis. University of Washington, Seattle, USA.

A112842 breaks down as A112843^2 times A112844, where A112844 is not necessarily squarefree.

3-pillows: A112833-A112835; 5-pillows: A112836-A112838; 7-pillows: A112839-A112841.

A112836 Number of domino tilings of a 5-pillow of order n.

Original entry on oeis.org

1, 2, 5, 13, 52, 261, 1666, 14400, 159250, 2308545, 43718544, 1079620569, 34863330980, 1466458546176, 80646187346132, 5787269582487581, 541901038236234048, 66279540183479379277, 10578427028263503488000

Offset: 0

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005

nonn

A 5-pillow is a generalized Aztec pillow. The 5-pillow of order n is a rotationally-symmetric region. It has a 2 X 2n central band of squares and then steps up from this band with steps of 5 horizontal squares to every 1 vertical square and steps down with steps of 1 horizontal square to every 1 vertical square.

			The number of domino tilings of the 5-pillow of order 6 is 1666=7^2*34.

C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis. University of Washington, Seattle, USA.

A112836 can be decomposed as A112837^2 times A112838, where A112838 is not necessarily squarefree.

3-pillows: A112833-A112835; 7-pillows: A112839-A112841; 9-pillows: A112842-A112844.

A112838 Small-number statistic from the enumeration of domino tilings of a 5-pillow of order n.

Original entry on oeis.org

1, 2, 5, 13, 13, 29, 34, 100, 130, 305, 361, 881, 1145, 2906, 3557, 8669, 10693, 26893, 33680, 83360, 102800, 254565, 317165, 790037, 980237, 2428298, 3011265, 7483801, 9301217, 23092857, 28646722, 71093860

Offset: 0

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005

A 5-pillow is a generalized Aztec pillow. The 5-pillow of order n is a rotationally-symmetric region. It has a 2 X 2n central band of squares and then steps up from this band with steps of 5 horizontal squares to every 1 vertical square and steps down with steps of 1 horizontal square to every 1 vertical square.

Plotting A112838(n+2)/A112838(n) gives an intriguing damped sine curve.

			The number of domino tilings of the 5-pillow of order 6 is 1666=7^2*34. A112838(n)=34.

C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis. University of Washington, Seattle, USA.

A112833 breaks down as A112834^2 times A112835, where A112835 is not necessarily squarefree.

3-pillows: A112833-A112835; 7-pillows: A112839-A112841; 9-pillows: A112842-A112844.

A112834 Large-number statistic from the enumeration of domino tilings of a 3-pillow of order n.

Original entry on oeis.org

1, 1, 1, 2, 3, 8, 19, 76, 263, 1584, 8199, 73272, 566401, 7555072, 87000289, 1730799376, 29728075177, 881736342784, 22583659690665, 998900331837728, 38149790451459859, 2516220411436892160, 143302702816187031875

Offset: 0

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005

nonn

A 3-pillow is also called an Aztec pillow. The 3-pillow of order n is a rotationally-symmetric region. It has a 2 X 2n central band of squares and then steps up from this band with steps of 3 horizontal squares to every 1 vertical square and steps down with steps of 1 horizontal square to every 1 vertical square.

			The number of domino tilings of the 3-pillow of order 4 is 117=3^2*13. A112834(4)=3.

C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis. University of Washington, Seattle, USA.

A112833 breaks down as a(n)^2 times A112835, where A112835 is not necessarily squarefree.

5-pillows: A112836-A112838; 7-pillows: A112839-A112841; 9-pillows: A112842-A112844.

A112840 Large-number statistic from the enumeration of domino tilings of a 7-pillow of order n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 7, 11, 28, 51, 154, 389, 1556, 4833, 22477, 80532, 440512, 1916580, 13388593, 73763989, 632754664, 4175659899, 42606281476, 336819337955, 4181786155008, 40981322633555, 630857431556758, 7576627032674784

Offset: 0

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005

nonn

A 7-pillow is a generalized Aztec pillow. The 7-pillow of order n is a rotationally-symmetric region. It has a 2 X 2n central band of squares and then steps up from this band with steps of 7 horizontal squares to every 1 vertical square and steps down with steps of 1 horizontal square to every 1 vertical square.

			The number of domino tilings of the 7-pillow of order 8 is 23353=11^2*193. A112840(n)=11.

C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis. University of Washington, Seattle, USA.

A112839 breaks down as A112840^2 times A112841, where A112841 is not necessarily squarefree.

3-pillows: A112833-A112835; 5-pillows: A112836-A112838; 9-pillows: A112842-A112844.

A112843 Large-number statistic from the enumeration of domino tilings of a 9-pillow of order n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 7, 11, 26, 44, 118, 221, 677, 1721, 6884, 21165, 95800, 324693, 1633462, 6253408, 35917622, 161554715, 1151376732, 6387653627, 54325024024, 348582834189, 3376194023305, 24664208882500, 273518249356480

Offset: 0

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005

nonn

A 9-pillow is a generalized Aztec pillow. The 9-pillow of order n is a rotationally-symmetric region. It has a 2 X 2n central band of squares and then steps up from this band with steps of 9 horizontal squares to every 1 vertical square and steps down with steps of 1 horizontal square to every 1 vertical square.

			The number of domino tilings of the 9-pillow of order 8 is 9065=7^2*185. A112843(n)=7.

C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis. University of Washington, Seattle, USA.

A112842 breaks down as A112843^2 times A112844, where A112844 is not necessarily squarefree.

3-pillows: A112833-A112835; 5-pillows: A112836-A112838; 7-pillows: A112839-A112841.

cp's OEIS Frontend

A112844 Small-number statistic from the enumeration of domino tilings of a 9-pillow of order n.

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A112833 Number of domino tilings of a 3-pillow of order n.

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A112835 Small-number statistic from the enumeration of domino tilings of a 3-pillow of order n.

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A112839 Number of domino tilings of a 7-pillow of order n.

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A112842 Number of domino tilings of a 9-pillow of order n.

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A112836 Number of domino tilings of a 5-pillow of order n.

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A112838 Small-number statistic from the enumeration of domino tilings of a 5-pillow of order n.

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A112834 Large-number statistic from the enumeration of domino tilings of a 3-pillow of order n.

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A112840 Large-number statistic from the enumeration of domino tilings of a 7-pillow of order n.

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