OFFSET

1,2

COMMENTS

For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.

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Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The ten 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 or d mod m)", (as in the relevant Mathematica programs):

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If f(n)=(n mod 3), then (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n); so that for n>=1, "(a, b, c, d mod m)" is given by

a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).

LINKS

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

EXAMPLE

Northwest corner:

1....3....5....8....11

2....4....6....9....13

7....10...14...19...25

12...16...21...28...36

17...23...30...39...50

22...29...38...49...63

MATHEMATICA

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

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

a=3; b=4; c2=5; d=6; m[n_]:=If[Mod[n, 4]==0, 1, 0];

f[n_]:=a*m[n+3]+b*m[n+2]+c2*m[n+1]+d*m[n]+5*Floor[(n-1)/4]

Table[f[n], {n, 1, 30}] (* A047207 *)

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}]] (* A191709 *)

Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191709 *)

CROSSREFS

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jun 12 2011

STATUS

approved