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 A114292 Modified Schroeder numbers for q=3. 8
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 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=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). 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 Alois P. Heinz, Rows n = 0..140, flattened EXAMPLE 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; MAPLE 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 MATHEMATICA b[x_, y_, k_] := b[x, y, k] = If[y>x || y

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Last modified April 20 06:36 EDT 2019. Contains 322294 sequences. (Running on oeis4.)