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A143734
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Number of paths of a generalized chess Queen from (0,0,0) to (n,n,n) in a cube, in which the Queen moves toward the goal point at each step.
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0
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1, 13, 638, 41476, 3015296, 232878412, 18691183682, 1540840801552, 129548309399618, 11057865563760844, 955237244106091682, 83324522236732005112, 7327068320498628273506, 648679579345635742189498
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| a(n) is the number of sequences whose terms are multiples of (0,0,1), (0,1,0), (1,0,0), (0,1,1), (1,0,1), (1,1,0), or (1,1,1) and whose sum is (n,n,n).
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FORMULA
| q(1,1,1) = 1; q(1,1,2) = 1; q(1,2,1) = 1; q(1,1,2) = 1; q(i_,j,k) = Sum(q(x,j,k), {x,1,i-1}) + Sum(q(i,y,k), {y,1,j-1}] + Sum(q(i,j,z), {z,1,k-1}) + Sum(q(i-w,j-w,k), {w,1,Min(i,j)}) + Sum(q(i,j-w,k-w), {w,1,Min(j, k)}) + Sum(q(i-w,j,k-w), {w,1,Min(i,k)}) + Sum(q(i-w,j-w,k-w), {w,1,Min(i,j,k)}); a(n) = q(n,n,n)
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EXAMPLE
| a(1)=13 because there are 13 generalized Queen paths from (0,0,0) to (1,1,1).
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MATHEMATICA
| q[1, 1, 1] = 1; q[1, 1, 2] = 1; q[1, 2, 1] = 1; q[1, 1, 2] = 1; q[i_, j_, k_] := q[i, j, k] = Sum[q[x, j, k], {x, 1, i - 1}] + Sum[q[i, y, k], {y, 1, j - 1}] + Sum[q[i, j, z], {z, 1, k - 1}] + Sum[q[i - w, j - w, k], {w, 1, Min[i, j]}] + Sum[q[i, j - w, k - w], {w, 1, Min[j, k]}] + Sum[q[i - w, j, k - w], {w, 1, Min[i, k]}] + Sum[q[i - w, j - w, k - w], {w, 1, Min[i, j, k]}]; a[n_] := q[n, n, n];
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CROSSREFS
| A132595 gives the two-dimensional version of this sequence.
Sequence in context: A142210 A109875 A067407 * A068232 A092547 A157027
Adjacent sequences: A143731 A143732 A143733 * A143735 A143736 A143737
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KEYWORD
| nonn
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AUTHOR
| Martin J. Erickson (erickson(AT)truman.edu), Aug 30 2008
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