cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

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

Original entry on oeis.org

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

Views

Author

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

Keywords

Comments

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.

Examples

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

References

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

Crossrefs

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

Views

Author

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

Keywords

Comments

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

Examples

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

Links

Crossrefs

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.

Programs

  • Maple
    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
  • Mathematica
    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

Views

Author

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

Keywords

Comments

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.

Examples

			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.

References

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

Links

Crossrefs

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.

Programs

  • PARI
    {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 */

Formula

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

Views

Author

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

Keywords

Comments

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.

Examples

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

References

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

Crossrefs

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.

A112841 Small-number statistic from the enumeration of domino tilings of a 7-pillow of order n.

Original entry on oeis.org

1, 2, 5, 13, 34, 34, 74, 73, 193, 256, 793, 1049, 2465, 2857, 6577, 8226, 21348, 28872, 74740, 91970, 222217, 268769, 669265, 852305, 2201945, 2805760, 7000777, 8636081, 21311098, 26588770, 67091170, 85150213
Offset: 0

Views

Author

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

Keywords

Comments

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.
Plotting A112841(n+2)/A112841(n) gives an intriguing damped sine curve.

Examples

			The number of domino tilings of the 7-pillow of order 8 is 23353=11^2*193. A112841(n)=193.

References

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

Crossrefs

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.

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

Views

Author

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

Keywords

Comments

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.

Examples

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

References

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

Crossrefs

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

Views

Author

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

Keywords

Comments

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.

Examples

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

References

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

Crossrefs

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

Views

Author

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

Keywords

Comments

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.

Examples

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

References

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

Crossrefs

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

Views

Author

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

Keywords

Comments

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.

Examples

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

References

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

Crossrefs

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

Views

Author

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

Keywords

Comments

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.

Examples

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

References

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

Crossrefs

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.
Showing 1-10 of 12 results. Next

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