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A131619 A general two modulo Ackermann recursion at 6 and 5. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..55.

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

Cf. A001695, A014221.

Sequence in context: A036012 A084401 A236465 * A048485 A127714 A283763

Adjacent sequences:  A131616 A131617 A131618 * A131620 A131621 A131622

KEYWORD

nonn,tabl,uned

AUTHOR

Roger L. Bagula, Oct 02 2007

STATUS

approved

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Last modified May 26 10:17 EDT 2022. Contains 354086 sequences. (Running on oeis4.)