A191665 Dispersion of A042963 (numbers >1, congruent to 1 or 2 mod 4), by antidiagonals.

1, 2, 3, 5, 6, 4, 10, 13, 9, 7, 21, 26, 18, 14, 8, 42, 53, 37, 29, 17, 11, 85, 106, 74, 58, 34, 22, 12, 170, 213, 149, 117, 69, 45, 25, 15, 341, 426, 298, 234, 138, 90, 50, 30, 16, 682, 853, 597, 469, 277, 181, 101, 61, 33, 19, 1365, 1706, 1194, 938, 554

Offset: 1

Author

Clark Kimberling, Jun 11 2011

Keywords

• nonn
• tabl

Comments

Row 1: A000975

Row 2: A081254

Row 3: A081253

Row 4: A052997

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:

...

A191663=dispersion of A042948 (0 or 1 mod 4 and >1)

A054582=dispersion of A005843 (0 or 2 mod 4 and >1; evens)

A191664=dispersion of A014601 (0 or 3 mod 4 and >1)

A191665=dispersion of A042963 (1 or 2 mod 4 and >1)

A191448=dispersion of A005408 (1 or 3 mod 4 and >1, odds)

A191666=dispersion of A042964 (2 or 3 mod 4)

...

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

A191663 has 1st col A042964, all else A042948

A054582 has 1st col A005408, all else A005843

A191664 has 1st col A042963, all else A014601

A191665 has 1st col A014601, all else A042963

A191448 has 1st col A005843, all else A005408

A191666 has 1st col A042948, all else A042964

...

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.

Examples

			Northwest corner:
1...2...5....10...21
3...6...13...26...53
4...9...18...37...74
7...14..29...58...117
8...17..34...69...138
		

Links

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

Crossrefs

Cf. A042963, A014601, A191426, A191664.

Programs

Mathematica

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a = 2; b = 5; m[n_] := If[Mod[n, 2] == 0, 1, 0];
f[n_] := a*m[n + 1] + b*m[n] + 4*Floor[(n - 1)/2]
Table[f[n], {n, 1, 30}]  (* A042963: (2+4k,5+4k) *)
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}]]
(* A191665 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
(* A191665  *)