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

1,2

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.

Links

Alois P. Heinz, Rows n = 1..2048, flattened

C. Kimberling, Fractal sequences

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)

Example

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

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 *)

Crossrefs

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

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

Cf. A000027, A004526, A002265, A132292.

Sequence in context: A233772 A056538 A266742 * A216477 A195836 A347285

Adjacent sequences: A120382 A120383 A120384 * A120386 A120387 A120388

Keyword

nonn,tabf

Author

Franklin T. Adams-Watters, Jun 29 2006

Status

approved