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 A143734 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. 1
 1, 13, 638, 41476, 3015296, 232878412, 18691183682, 1540840801552, 129548309399618, 11057865563760844, 955237244106091682, 83324522236732005112, 7327068320498628273506, 648679579345635742189498, 57761885964038080406607410, 5169168679056263697679753150 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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). LINKS Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..250 (first 81 terms from Alois P. Heinz) 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). a(n) ~ c * d^(3*n) / n, where d = 4.575760096729293131840036142861966071... is the root of the equation -8 - 11*d - 9*d^2 - 2*d^3 + d^4 = 0, and c = 0.14917103190900041974882341373298677... . - Vaclav Kotesovec, Aug 23 2014 EXAMPLE a(1)=13 because there are 13 generalized Queen paths from (0,0,0) to (1,1,1). MAPLE b:= proc(x, y, z) option remember; `if`(x=0 and y=0 and z=0, 1,       add(b(x-i, y, z), i=1..x)+ add(b(x, y-i, z), i=1..y)+       add(b(x, y, z-i), i=1..z)+ add(b(x-i, y-i, z), i=1..min(x, y))+       add(b(x-i, y, z-i), i=1..min(x, z))+ add(b(x, y-i, z-i),       i=1..min(y, z))+ add(b(x-i, y-i, z-i), i=1..min(x, y, z)))     end: a:= n-> b(n\$3): seq(a(n), n=0..20);  # Alois P. Heinz, Jul 23 2012 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]; CROSSREFS A132595 gives the two-dimensional version of this sequence. Sequence in context: A296951 A351507 A067407 * A316331 A217380 A068232 Adjacent sequences:  A143731 A143732 A143733 * A143735 A143736 A143737 KEYWORD nonn AUTHOR Martin J. Erickson (erickson(AT)truman.edu), Aug 30 2008 STATUS approved

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Last modified August 12 04:52 EDT 2022. Contains 356067 sequences. (Running on oeis4.)