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.

Previous Showing 11-19 of 19 results.

A114296 First row of Modified Schroeder numbers for q=3 (A114292).

Original entry on oeis.org

1, 1, 2, 5, 16, 57, 224, 934, 4092, 18581, 86888, 415856, 2029160, 10061161, 50568680, 257129888, 1320619176, 6842177174, 35722456976, 187772944964, 992991472328, 5279633960181, 28208037066528, 151373637844440, 815568695756496, 4410124252008112
Offset: 0

Views

Author

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

Keywords

Comments

a(i) is the number of paths from (0,0) to (i,i) using steps of length (0,1), (1,0) and (1,1), not passing above the line y=x nor below the line y=x/2.

Examples

			The number of paths from (0,0) to (3,3) staying between the lines y=x and y=x/2 using steps of length (0,1), (1,0) and (1,1) is a(3)=5.

Links

Crossrefs

See also A112833-A112844 and A114292-A114299.
Cf. A224776, A225041. - Alois P. Heinz, Apr 25 2013
Cf. A286761.

Programs

  • Maple
    b:= proc(x, y) option remember; `if`(y>x or y b(n, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 25 2013
  • Mathematica
    b[x_, y_] := b[x, y] = If[y>x || yJean-François Alcover, Feb 05 2015, after Alois P. Heinz *)

Formula

a(n) ~ c * (3+2*sqrt(2))^n / n^(3/2), where c = 0.02741316010407391604887680145773... . - Vaclav Kotesovec, Sep 07 2014

Extensions

Corrected by Philippe Deléham, Sep 04 2006
Extended beyond a(10) by Alois P. Heinz, Apr 25 2013

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.

A112837 Large-number statistic from the enumeration of domino tilings of a 5-pillow of order n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 7, 12, 35, 87, 348, 1107, 5518, 22464, 150574, 817057, 7118856, 49644383, 560434040, 5142118400, 76370120248, 914476059335, 17638655014128, 274908897964359, 6936239946318204, 141510942505315328
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. A112837(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

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.

A114293 Modified Schroeder numbers for q=5.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 5, 5, 5, 2, 1, 13, 13, 13, 5, 2, 1, 42, 42, 42, 16, 6, 2, 1, 150, 150, 150, 57, 21, 6, 2, 1, 553, 553, 553, 210, 77, 21, 6, 2, 1, 2202, 2202, 2202, 836, 306, 82, 22, 6, 2, 1, 9233, 9233, 9233, 3505, 1282, 341, 89, 22, 6, 2, 1, 39726, 39726, 39726
Offset: 0

Views

Author

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

Keywords

Comments

a(i,j) is the number of paths from (i,i) to (j,j) using steps of length (0,1), (1,0) and (1,1), not passing above the line y=x nor below the line y=2x/3. The Hamburger Theorem implies that we can use this table to calculate the number of domino tilings of an Aztec 5-pillow (A112836). To calculate this quantity, let P_n = the principal n X n submatrix of this array. If J_n = the back-diagonal matrix of order n, then A112836(n)=det(P_n+J_nP_n^(-1)J_n).

Examples

			The number of paths from (0,0) to (4,4) staying between the lines y=x and y=2x/3 using steps of length (0,1), (1,0) and (1,1) is a(0,4)=5.

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

See also A112833-A112844 and A114292-A114299.

A114294 Modified Schroeder numbers for q=7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 5, 5, 5, 5, 2, 1, 13, 13, 13, 13, 5, 2, 1, 34, 34, 34, 34, 13, 5, 2, 1, 110, 110, 110, 110, 42, 16, 6, 2, 1, 393, 393, 393, 393, 150, 57, 21, 6, 2, 1, 1449, 1449, 1449, 1449, 553, 210, 77, 21, 6, 2, 1, 5390, 5390, 5390, 5390, 2057, 781
Offset: 0

Views

Author

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

Keywords

Comments

a(i,j) is the number of paths from (i,i) to (j,j) using steps of length (0,1), (1,0) and (1,1), not passing above the line y=x nor below the line y=3x/4. The Hamburger Theorem implies that we can use this table to calculate the number of domino tilings of an Aztec 7-pillow (A112839). To calculate this quantity, let P_n = the principal n X n submatrix of this array. If J_n = the back-diagonal matrix of order n, then A112839(n)=det(P_n+J_nP_n^(-1)J_n).

Examples

			The number of paths from (0,0) to (5,5) staying between the lines y=x and y=3x/4 using steps of length (0,1), (1,0) and (1,1) is a(0,5)=5.

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

See also A112833-A112844 and A114292-A114299.

A114295 Modified Schroeder numbers for q=9.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 5, 5, 5, 5, 5, 2, 1, 13, 13, 13, 13, 13, 5, 2, 1, 34, 34, 34, 34, 34, 13, 5, 2, 1, 89, 89, 89, 89, 89, 34, 13, 5, 2, 1, 288, 288, 288, 288, 288, 110, 42, 16, 6, 2, 1, 1029, 1029, 1029, 1029, 1029, 393, 150, 57, 21, 6, 2, 1
Offset: 0

Views

Author

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

Keywords

Comments

a(i,j) is the number of paths from (i,i) to (j,j) using steps of length (0,1), (1,0) and (1,1), not passing above the line y=x nor below the line y=4x/5. The Hamburger Theorem implies that we can use this table to calculate the number of domino tilings of an Aztec 9-pillow (A112842). To calculate this quantity, let P_n = the principal n X n submatrix of this array. If J_n = the back-diagonal matrix of order n, then A112842(n)=det(P_n+J_nP_n^(-1)J_n).

Examples

			The number of paths from (0,0) to (6,6) staying between the lines y=x and y=4x/5 using steps of length (0,1), (1,0) and (1,1) is a(0,6)=5.

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

See also A112833-A112844 and A114292-A114299.

A114298 First row of Modified Schroeder numbers for q=7 (A114294).

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 13, 34, 110, 393, 1449, 5390, 21534, 90418, 389265, 1694769, 7593330, 34910142, 163314286, 772044618, 3702870682, 18017064221, 88689351909, 440271808570, 2205020557614, 11141413883818, 56737939027682
Offset: 0

Views

Author

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

Keywords

Comments

a(i) is the number of paths from (0,0) to (i,i) using steps of length (0,1), (1,0) and (1,1), not passing above the line y=x nor below the line y=3x/4.

Examples

			The number of paths from (0,0) to (5,5) staying between the lines y=x and y=3x/4 using steps of length (0,1), (1,0) and (1,1) is a(5)=5.

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

See also A112833-A112844 and A114292-A114299.

A114297 First row of Modified Schroeder numbers for q=5 (A114293).

Original entry on oeis.org

1, 1, 1, 2, 5, 13, 42, 150, 553, 2202, 9233, 39726, 176932, 810798, 3786137, 18022100, 87265298, 428202617, 2127088358, 10684752474, 54181245592, 277101480826, 1428262595206, 7412626391101, 38712130945272, 203330779196084
Offset: 0

Views

Author

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

Keywords

Comments

a(i) is the number of paths from (0,0) to (i,i) using steps of length (0,1), (1,0) and (1,1), not passing above the line y=x nor below the line y=2x/3.

Examples

			The number of paths from (0,0) to (4,4) staying between the lines y=x and y=2x/3 using steps of length (0,1), (1,0) and (1,1) is a(4)=5.

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

See also A112833-A112844 and A114292-A114299.
Previous Showing 11-19 of 19 results.

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