OFFSET

1,2

COMMENTS

For a background discussion of dispersions, see A191426.

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Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The six sequences and dispersions are listed here:

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EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):

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Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c mod m)", (as in the Mathematica program below):

If f(n)=(n mod 3), then (a,b,c,a,b,c,a,b,c,...) is given by

a*f(n+2)+b*f(n+1)+c*f(n), so that "(a or b or c mod m)" is given by

a*f(n+2)+b*f(n+1)+c*f(n)+m*floor((n-1)/3)), for n>=1.

LINKS

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

EXAMPLE

Northwest corner:

1....2....3....5....7

4....6....9....13...18

8....11...15...21...29

12...17...23...31...42

16...22...30...41...55

MATHEMATICA

(* Program generates the dispersion array T of the increasing sequence f[n] *)

r = 40; r1 = 12; c = 40; c1 = 12;

a = 2; b = 3; c2 = 5; m[n_] := If[Mod[n, 3] == 0, 1, 0];

f[n_] := a*m[n + 2] + b*m[n + 1] + c2*m[n] + 4*Floor[(n - 1)/3]

Table[f[n], {n, 1, 30}] (* A042968 *)

mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]

rows = {NestList[f, 1, c]};

Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];

t[i_, j_] := rows[[i, j]];

TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191670 *)

Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191670 *)

CROSSREFS

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jun 11 2011

STATUS

approved