A181631 Triangle by rows, number of leading 1's in the maximal Fibonacci representation (A104326).

1, 1, 2, 1, 2, 3, 1, 1, 2, 3, 4, 1, 1, 1, 2, 2, 3, 4, 5, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 6, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 7, 8

Offset: 1

Author

Gary W. Adamson, Nov 02 2010

Keywords

• nonn
• tabf

Comments

Row sums = A001911: (1, 3, 6, 11, 19, 32, ...).

A112310 = number of 1's in the maximal Fibonacci representation, which has headings of (..., 8, 5, 3, 2, 1) filling entries from the right to left; as opposed to the minimal Fibonacci representation (A014417) which starts from the left. For example, 8 in maximal = 1011 = (5 + 2 + 1) whereas in minimal = (10000) = (8).

Rows have (1, 2, 3, 5, 8, ...) terms.

Examples

			First few rows of the triangle:
  1;
  1, 2;
  1, 2, 3;
  1, 1, 2, 3, 4;
  1, 1, 1, 2, 2, 3, 4, 5;
  1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 6;
  ...
Example: a(14) = 1 since 14 in the maximal Fibonacci representation is 10111.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000045 (row lengths), A003754, A001911 (row sums), A014417, A090996, A104326, A112310.

Programs

Mathematica

f[s_] := Module[{i = FirstPosition[s, 0]}, If[MissingQ[i], Length[s], i[[1]] - 1]]; f /@ Select[IntegerDigits[#, 2] & /@ Range[300], SequencePosition[#, {0, 0}] == {} &] (* Amiram Eldar, May 31 2025 *)

Formula

a(n) = A090996(A003754(n+1)). - Amiram Eldar, May 31 2025

Extensions

More terms from Amiram Eldar, May 31 2025