OFFSET
1,2
COMMENTS
This double modulo Ackermann function was inspired by the tiling problem given in "Elements of the Theory of Computation" which resembles an Ackermann recursion. The {a,b}->{5,6} was designed for the 10 X 10 output given to be active. Without the modulo this function is effectively limited to 4 X 4 in Mathematica by computation time.
REFERENCES
S. Wolfram, A New Kind of Science. Champaign, IL: Wolfram Media, p. 906, 2002.
Harry R. Lewis and Christos H. Papadimitriou, Elements of the Theory of Computation, Prentice-Hall, 1981, page 296 and 345.
LINKS
Eric Weisstein's World of Mathematics, Ackermann function.
FORMULA
a(1, n) = n mod 6; a(m, 1) = a(m - 1, 2); a(m, n) = a(m - 1, a(m, n - 1) + 1) mod 5.
aout(n,m) = AntidiagonalTransform(a(n,m)).
EXAMPLE
{1},
{2, 2},
{3, 3, 3},
{0, 0, 4, 4},
{3, 3, 2, 0, 5},
{1, 1, 4, 4, 1, 0},
{1, 1, 3, 1, 1, 2, 1},
{1, 1, 1, 1, 0, 3, 3, 2},
{1, 1, 1, 1, 3, 3, 0, 4, 3},
{1, 1, 1, 1, 1, 1, 4, 2, 0, 4}
MATHEMATICA
f[1, n_] := Mod[n, 6]; f[m_, 1] := f[m - 1, 2]; f[m_, n_] := Mod[f[m - 1, f[m, n - 1] + 1], 5];
a0 = Table[f[a, b], {a, 1, 10}, {b, 1, 10}];
ListDensityPlot[%, ColorFunction -> (Hue[2# ] &)];
Dimensions[a0];
(* antidiagonal transform*)
c = Delete[Table[Reverse[Table[a0[[n, l - n]], {n, 1, l - 1}]], {l, 1, Dimensions[a0][[1]] + 1}], 1];
Flatten[c]
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Oct 02 2007
STATUS
approved