A114299 -id:A114299 - cp's OEIS frontend

A114292 Modified Schroeder numbers for q=3.

1, 1, 1, 2, 2, 1, 5, 5, 2, 1, 16, 16, 6, 2, 1, 57, 57, 21, 6, 2, 1, 224, 224, 82, 22, 6, 2, 1, 934, 934, 341, 89, 22, 6, 2, 1, 4092, 4092, 1492, 384, 90, 22, 6, 2, 1, 18581, 18581, 6770, 1729, 393, 90, 22, 6, 2, 1, 86888, 86888, 31644, 8044, 1794, 394, 90, 22, 6, 2, 1

Offset: 0

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

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=x/2. The Hamburger Theorem implies that we can use this table to calculate the number of domino tilings of an Aztec 3-pillow (A112833). 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 A112833(n)=det(P_n+J_nP_n^(-1)J_n).

			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(0,3)=5.
Triangle begins:
1;
1, 1;
2, 2, 1;
5, 5, 2, 1;
16, 16, 6, 2, 1;
57, 57, 21, 6, 2, 1;
224, 224, 82, 22, 6, 2, 1;
934, 934, 341, 89, 22, 6, 2, 1;
4092, 4092, 1492, 384, 90, 22, 6, 2, 1;

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

Alois P. Heinz, Rows n = 0..140, flattened

See also A112833-A112844 and A114293-A114299.

b:= proc(x, y, k) option remember;
      `if`(y>x or y b(n, n, k):
seq(seq(a(n,k), k=0..n), n=0..12); # Alois P. Heinz, Apr 26 2013

b[x_, y_, k_] := b[x, y, k] = If[y>x || yJean-François Alcover, Mar 06 2015, after Alois P. Heinz *)

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

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

nonn

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.

			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.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
C. Hanusa, A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows, PhD Thesis, 2005, University of Washington, Seattle, USA.

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

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

b[x_, y_] := b[x, y] = If[y>x || yJean-François Alcover, Feb 05 2015, after Alois P. Heinz *)

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

Corrected by Philippe Deléham, Sep 04 2006

Extended beyond a(10) by Alois P. Heinz, Apr 25 2013

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

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

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).

			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.

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

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

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

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).

			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.

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

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

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

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).

			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.

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

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

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

nonn

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.

			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.

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

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

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

nonn

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.

			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.

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

See also A112833-A112844 and A114292-A114299.

cp's OEIS Frontend

A114292 Modified Schroeder numbers for q=3.

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

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A114293 Modified Schroeder numbers for q=5.

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A114294 Modified Schroeder numbers for q=7.

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A114295 Modified Schroeder numbers for q=9.

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

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

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