A120385 If a(n-1) = 1 then largest value so far + 1, otherwise floor(a(n-1)/2); or table T(n,k) with T(n,0) = n, T(n,k+1) = floor(T(n,k)/2).

1, 2, 1, 3, 1, 4, 2, 1, 5, 2, 1, 6, 3, 1, 7, 3, 1, 8, 4, 2, 1, 9, 4, 2, 1, 10, 5, 2, 1, 11, 5, 2, 1, 12, 6, 3, 1, 13, 6, 3, 1, 14, 7, 3, 1, 15, 7, 3, 1, 16, 8, 4, 2, 1, 17, 8, 4, 2, 1, 18, 9, 4, 2, 1, 19, 9, 4, 2, 1, 20, 10, 5, 2, 1, 21, 10, 5, 2, 1, 22, 11, 5, 2, 1, 23, 11, 5, 2, 1, 24, 12, 6, 3, 1, 25

Offset: 1

Author

Franklin T. Adams-Watters, Jun 29 2006

Keywords

• nonn
• tabf

Comments

Although not strictly a fractal sequence as defined in the Kimberling link, this sequence has many fractal properties. If the first instance of each value is removed, the result is the original sequence with each row repeated twice. Removing all odd-indexed instances of each value does give the original sequence.

Examples

			The table starts:
  1;
  2, 1;
  3, 1;
  4, 2, 1;
  5, 2, 1;
  6, 3, 1;
  7, 3, 1;
  8, 4, 2, 1;
		

Links

• Alois P. Heinz, Rows n = 1..2048, flattened
• C. Kimberling, Fractal sequences

Crossrefs

Cf. A029837 (row lengths), A083652 (position of first n).

Cf. A005187 (row sums). A001477, A050292, A080277.

Cf. A000027, A004526, A002265, A132292.

Programs

Maple

T:= proc(n) T(n):= `if`(n=1, 1, [n, T(iquo(n, 2))][]) end:
seq(T(n), n=1..30);  # Alois P. Heinz, Feb 12 2019

Mathematica

Flatten[Function[n,NestWhile[Append[#, Floor[Last[#]/2]] &, {n}, Last[#] != 1 &]][#] & /@ Range[50]] (* Birkas Gyorgy, Apr 14 2011 *)

Formula

T(n,k) = floor(n/2^(k-1)).

From Peter Bala, Feb 02 2013: (Start)

The n-th row polynomial R(n,t) = Sum_{k>=0} t^k*floor(n/2^k) and satisfies the recurrence equation R(n,t) = t*R(floor(n/2),t) + n, with R(1,t) = 1.

O.g.f. Sum_{n>=1} R(n,t)*x^n = 1/(1-x)*Sum_{n>=0} t^n*x^(2^n)/(1 - x^(2^n)).

Product_{n>=1} ( 1 + x^((t^n - 2^n)/(t-2)) ) = 1 + Sum_{n>=1} x^R(n,t) = 1 + x + x^(2 + t) + x^(3 + t) + x^(4 + 2*t + t^2) + .... For related sequences see A050292 (t = -1), A001477(t = 0), A005187 (t = 1) and A080277 (t = 2).

(End)