A191455 Dispersion of (floor(n*e)), by antidiagonals.

1, 2, 3, 5, 8, 4, 13, 21, 10, 6, 35, 57, 27, 16, 7, 95, 154, 73, 43, 19, 9, 258, 418, 198, 116, 51, 24, 11, 701, 1136, 538, 315, 138, 65, 29, 12, 1905, 3087, 1462, 856, 375, 176, 78, 32, 14, 5178, 8391, 3974, 2326, 1019, 478, 212, 86, 38, 15, 14075, 22809

Offset: 1

Author

Clark Kimberling, Jun 05 2011

Keywords

• nonn
• tabl

Comments

Background discussion: Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1. The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...). Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers. The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence. Examples:

(1) s=A000040 (the primes), D=A114537, u=A114538.

(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.

(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.

More recent examples of dispersions: A191426-A191455.

Examples

			Northwest corner:
  1...2....5....13...35
  3...8....21...57...154
  4...10...27...73...198
  6...16...43...116..315
  7...19...51...138..375
		

Crossrefs

Cf. A114537, A035513, A035506.

Programs

Maple

A191455 := proc(r, c)
    option remember;
    if c = 1 then
        A054385(r) ;
    else
        A022843(procname(r, c-1)) ;
    end if;
end proc: # R. J. Mathar, Jan 25 2015

Mathematica

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;
f[n_] :=Floor[n*E]   (* complement of column 1 *)
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}]]
(* A191455 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191455 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)