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
KEYWORD
nonn
AUTHOR
Martin J. Erickson (erickson(AT)truman.edu), Aug 30 2008
STATUS
approved