OFFSET
1,2
COMMENTS
Row 1: A000225 (-1+2^n)
Row 2: A000079 (2^n)
Row 3: A055010
Row 4: 3*A000079
Row 5: A153894
Row 6: 5*A000079
Row 7: A086224
Row 8: A005009
Row 9: A052996
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:
...
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
...
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.
This sequence is a permutation of the natural numbers. - L. Edson Jeffery, Aug 13 2014
LINKS
Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)
EXAMPLE
Northwest corner:
1...3...7....15...31
2...4...8....16...32
5...11..23...47...95
6...12..24...48...96
9...19..39...79...159
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; 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}] (* A014601(n+2): (4+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}]] (* A191664 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191664 *)
(* Clark Kimberling, Jun 11 2011 *)
Grid[Table[2^k*(2*Floor[(n + 1)/2] - 1) - Mod[n, 2], {n, 12}, {k, 12}]] (* L. Edson Jeffery, Aug 13 2014 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jun 11 2011
STATUS
approved