A191735 Dispersion of A047223, (numbers >1 and congruent to 1 or 2 or 3 mod 5), by antidiagonals.

1, 2, 4, 3, 7, 5, 6, 12, 8, 9, 11, 21, 13, 16, 10, 18, 36, 22, 27, 17, 14, 31, 61, 37, 46, 28, 23, 15, 52, 102, 62, 77, 47, 38, 26, 19, 87, 171, 103, 128, 78, 63, 43, 32, 20, 146, 286, 172, 213, 131, 106, 72, 53, 33, 24, 243, 477, 287, 356, 218, 177, 121, 88

Offset

1,2

Comments

For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702.

...

Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}. Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5. There are 10 sequences in S, each matched by a (nearly) complementary sequence in T. Each of the 20 sequences generates a dispersion, as listed here:

...

A191722=dispersion of A008851 (0, 1 mod 5 and >1)

A191723=dispersion of A047215 (0, 2 mod 5 and >1)

A191724=dispersion of A047218 (0, 3 mod 5 and >1)

A191725=dispersion of A047208 (0, 4 mod 5 and >1)

A191726=dispersion of A047216 (1, 2 mod 5 and >1)

A191727=dispersion of A047219 (1, 3 mod 5 and >1)

A191728=dispersion of A047209 (1, 4 mod 5 and >1)

A191729=dispersion of A047221 (2, 3 mod 5 and >1)

A191730=dispersion of A047211 (2, 4 mod 5 and >1)

A191731=dispersion of A047204 (3, 4 mod 5 and >1)

...

A191732=dispersion of A047202 (2,3,4 mod 5 and >1)

A191733=dispersion of A047206 (1,3,4 mod 5 and >1)

A191734=dispersion of A032793 (1,2,4 mod 5 and >1)

A191735=dispersion of A047223 (1,2,3 mod 5 and >1)

A191736=dispersion of A047205 (0,3,4 mod 5 and >1)

A191737=dispersion of A047212 (0,2,4 mod 5 and >1)

A191738=dispersion of A047222 (0,2,3 mod 5 and >1)

A191739=dispersion of A008854 (0,1,4 mod 5 and >1)

A191740=dispersion of A047220 (0,1,3 mod 5 and >1)

A191741=dispersion of A047217 (0,1,2 mod 5 and >1)

...

For further information about these 20 dispersions, see A191722.

...

Regarding the dispersions A191722-A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.

Links

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

Example

Northwest corner:
1....2....3....6....11
4....7....12...21...36
5....8....13...22...37
9....16...27...46...77
10...17...28...47...78
14...23...38...63...106

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=6; m[n_]:=If[Mod[n, 3]==0, 1, 0];
f[n_]:=a*m[n+2]+b*m[n+1]+c2*m[n]+5*Floor[(n-1)/3]
Table[f[n], {n, 1, 30}]  (* A047223 *)
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}]] (* A191735 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191735  *)

Crossrefs

Cf. A047208, A047223, A191722, , A191725, A191426.

Sequence in context: A194031 A340245 A064357 * A376133 A191666 A215673

Adjacent sequences: A191732 A191733 A191734 * A191736 A191737 A191738

Keyword

nonn,tabl

Author

Clark Kimberling, Jun 14 2011

Status

approved