_Clark Kimberling (ck6(AT)evansville.edu), _, Jun 10 2011

proposed

approved

Row 1: A152009.

A191654A191450=dispersion of A016789 (2 mod 3)

A191654 A191450 has 1st col A032766, all else A016789

5...9...15...24...37.

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

allocated for Clark Kimberling

Dispersion of (2,5,8,11,14,17,...), by antidiagonals.

1, 3, 2, 6, 4, 5, 10, 7, 9, 8, 16, 12, 15, 13, 11, 25, 19, 24, 21, 18, 14, 39, 30, 37, 33, 28, 22, 17, 60, 46, 57, 51, 43, 34, 27, 20, 91, 70, 87, 78, 66, 52, 42, 31, 23, 138, 106, 132, 118, 100, 79, 64, 48, 36, 26, 208, 160, 199, 178, 151, 120, 97, 73, 55

1,2

For a background discussion of dispersions, see A191426.

...

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:

...

A191449=dispersion of A008583 (0 mod 3)

A191451=dispersion of A016777 (1 mod 3)

A191654=dispersion of A016789 (2 mod 3)

A191656=dispersion of A001651 (1 or 2 mod 3)

A083044=dispersion of A007494 (0 or 2 mod 3)

A191655=dispersion of A032766 (0 or 1 mod 3)

...

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

A191449 has 1st col A001651, all else A008583

A191451 has 1st col A007494, all else A016777

A191654 has 1st col A032766, all else A016789

A191656 has 1st col A008583, all else A001651

A083044 has 1st col A016777, all else A083044

A191655 has 1st col A016789, all else A032766

...

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.

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

(* 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 *)

Cf. A016789, A032766, A191426.

allocated

nonn,tabl

Clark Kimberling (ck6(AT)evansville.edu), Jun 10 2011

approved

proposed

allocated for Clark Kimberling

allocated

approved