A123390 Triangle read by rows: n-th row starts with n and continues with half the previous value as long as that is even.

1, 2, 1, 3, 4, 2, 1, 5, 6, 3, 7, 8, 4, 2, 1, 9, 10, 5, 11, 12, 6, 3, 13, 14, 7, 15, 16, 8, 4, 2, 1, 17, 18, 9, 19, 20, 10, 5, 21, 22, 11, 23, 24, 12, 6, 3, 25, 26, 13, 27, 28, 14, 7, 29, 30, 15, 31, 32, 16, 8, 4, 2, 1, 33, 34, 17, 35, 36, 18, 9, 37, 38, 19, 39, 40, 20, 10, 5, 41, 42, 21

Offset: 1

Author

Franklin T. Adams-Watters, Oct 13 2006

Keywords

• nonn
• tabf

Comments

A fractal sequence, generated by the rule a(n) is a new maximum when a(n-1) is odd and a repetition of an earlier value when a(n-1) is even.

From Flávio V. Fernandes, Mar 13 2025: (Start)

a(n) is given by A003602(n) at A001511(n) diagram

1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9

. 1 . . . 2 . . . 3 . . . 4 . . .

. . . 1 . . . . . . . 2 . . . . .

. . . . . . . 1 . . . . . . . . .

. . . . . . . . . . . . . . . 1 .

read by backwards 2^n, which is given by A118319(n) at A001511(n) diagram

1 . 2 . 4 . 5 . 8 . 9 .11 .12 .16

. 3 . . . 6 . . .10 . . .13 . . .

. . . 7 . . . . . . .14 . . . . .

. . . . . . .15 . . . . . . . . .

. . . . . . . . . . . . . . .31 . - see formula. (End)

Examples

			Triangle starts
  1;
  2, 1;
  3;
  4, 2, 1;
  5;
  6, 3;
  7;
  8, 4, 2, 1;
  9;
  10, 5;
  11;
  12, 6, 3;
  13;
		

Links

• Alois P. Heinz, Rows n = 1..10000, flattened

Crossrefs

Row lengths are A001511.

Row sums give A129527.

Cf. A120385.

Cf. A003602, A108918, A118319.

Programs

Maple

T:= proc(n) local m,l; m:=n; l:= m;
      while irem(m, 2, 'm')=0 do l:=l,m od: l
    end:
seq(T(n), n=1..40);  # Alois P. Heinz, Oct 09 2015

Mathematica

Flatten[Function[n, NestWhile[Append[#, Last[#]/2] &, {n}, EvenQ[Last[#]] &]][#] & /@ Range[20]] (* Birkas Gyorgy, Apr 13 2011 *)

Formula

a(1) = 1, for n > 1, if a(n-1) is even, a(n) = a(n-1)/2, otherwise a(n) = (max_{k

Ordinal transform of A082850.

a(n) = A003602(A108918(n)). - Flávio V. Fernandes, Mar 13 2025

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