A191437 Dispersion of ([n*x+n+x-2]), where x=(golden ratio) and [ ]=floor, by antidiagonals.

1, 3, 2, 8, 5, 4, 21, 13, 11, 6, 55, 34, 29, 16, 7, 144, 89, 76, 42, 18, 9, 377, 233, 199, 110, 47, 24, 10, 987, 610, 521, 288, 123, 63, 26, 12, 2584, 1597, 1364, 754, 322, 165, 68, 32, 14, 6765, 4181, 3571, 1974, 843, 432, 178, 84, 37, 15, 17711, 10946

Offset: 1

Author

Clark Kimberling, Jun 04 2011

Keywords

• nonn
• tabl

Comments

Rows 1 and 2: Fibonacci numbers. Rows 3 and 5: Lucas numbers. Row n satisfies the recurrence x(n)=3*x(n-1)-x(n-2).

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....3....8....21...55
  2....5....13...34...89
  4....11...29...76...199
  6....16...42...110..288
  7....18...47...123..322
		

Crossrefs

Cf. A114537, A035513, A035506, A000045, A000032, A191426.

Programs

Mathematica

(* Program generates the dispersion array T of increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;  x = GoldenRatio;
f[n_] := Floor[n*x+n+x-2] (* 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}]]
(* A191437 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191437 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)