OFFSET
1,2
COMMENTS
Row 1: A152009.
For a background discussion of dispersions, see A191426.
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Each of the sequences (3n, n>0), (3n+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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There is a formula for sequences of the type "(a or b mod m)", (as in the Mathematica program below):
If f(n)=(n mod 2), then (a,b,a,b,a,b,...) is given by
a*f(n+1)+b*f(n), so that "(a or b mod m)" is given by
a*f(n+1)+b*f(n)+m*floor((n-1)/2)), for n>=1.
EXAMPLE
Northwest corner:
1...3...6....10...16
2...4...7....12...19
5...9...15...24...37
8...13..21...33...51
11..18..28...43...66
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; m[n_] := If[Mod[n, 2] == 0, 1, 0];
f[n_] := a*m[n + 1] + b*m[n] + 3*Floor[(n - 1)/2]
Table[f[n], {n, 1, 30}] (* A032766: (3+5k, 4+5k, k>=0) *)
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}]] (* A191655 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191655 sequence *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jun 10 2011
STATUS
approved