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
KEYWORD
nonn,tabf
AUTHOR
Franklin T. Adams-Watters, Jun 29 2006
STATUS
approved