A191672 Dispersion of A042965 (>1 and congruent to 0 or 1 or 3 mod 4), by antidiagonals.

1, 3, 2, 5, 4, 6, 8, 7, 9, 10, 12, 11, 13, 15, 14, 17, 16, 19, 21, 20, 18, 24, 23, 27, 29, 28, 25, 22, 33, 32, 37, 40, 39, 35, 31, 26, 45, 44, 51, 55, 53, 48, 43, 36, 30, 61, 60, 69, 75, 72, 65, 59, 49, 41, 34, 83, 81, 93, 101, 97, 88, 80, 67, 56, 47, 38

Offset: 1

Author

Clark Kimberling, Jun 11 2011

Keywords

• nonn
• tabl

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:

...

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

...

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.

Examples

			Northwest corner:
1....3...5....8....12
2....4...7....11...16
6....9...13...19...27
10...15..21...29...40
14...20..28...39...53
		

Links

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

Crossrefs

Cf. A016825, A042965, A191668, A191452.

Programs

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; 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}] (* A042965 *)
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}]] (* A191672 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191672 *)