A191671 Dispersion of A004772 (>1 and congruent to 0 or 2 or 3 mod 4), by antidiagonals.

1, 2, 5, 3, 7, 9, 4, 10, 12, 13, 6, 14, 16, 18, 17, 8, 19, 22, 24, 23, 21, 11, 26, 30, 32, 31, 28, 25, 15, 35, 40, 43, 42, 38, 34, 29, 20, 47, 54, 58, 56, 51, 46, 39, 33, 27, 63, 72, 78, 75, 68, 62, 52, 44, 37, 36, 84, 96, 104, 100, 91, 83, 70, 59, 50, 41

Offset

1,2

Comments

For a background discussion of dispersions, see A191426.

...

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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A191452=dispersion of A008586 (4k, k>=1)

A191667=dispersion of A016813 (4k+1, k>=1)

A191668=dispersion of A016825 (4k+2, k>=0)

A191669=dispersion of A004767 (4k+3, k>=0)

A191670=dispersion of A042968 (1 or 2 or 3 mod 4 and >=2)

A191671=dispersion of A004772 (0 or 1 or 3 mod 4 and >=2)

A191672=dispersion of A004773 (0 or 1 or 2 mod 4 and >=2)

A191673=dispersion of A004773 (0 or 2 or 3 mod 4 and >=2)

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

A191452 has 1st col A042968, all else A008486

A191667 has 1st col A004772, all else A016813

A191668 has 1st col A042965, all else A016825

A191669 has 1st col A004773, all else A004767

A191670 has 1st col A008486, all else A042968

A191671 has 1st col A016813, all else A004772

A191672 has 1st col A016825, all else A042965

A191673 has 1st col A004767, all else A004773

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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....4....6
5....7....10...14...19
9....12...16...22...30
13...18...24...32...43
17...23...31...42...56

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 = 4; 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}]  (* A004772 *)
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}]] (* A191671 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191671  *)

Crossrefs

Cf. A016813, A004772, A191667, A191452.

Sequence in context: A272908 A191432 A135587 * A176707 A257533 A308021

Adjacent sequences: A191668 A191669 A191670 * A191672 A191673 A191674

Keyword

nonn,tabl

Author

Clark Kimberling, Jun 11 2011

Status

approved