A362781 Natural numbers n for which some base-phi representation of n is anti-palindromic.

0, 1, 3, 4, 5, 6, 8, 11, 13, 14, 15, 16, 21, 23, 29, 31, 33, 35, 37, 39, 41, 43, 45, 53, 55, 61, 63, 76, 78, 80, 86, 88, 89, 91, 97, 99, 100, 102, 108, 110, 111, 113, 119, 121, 136, 138, 144, 146, 158, 160, 166, 168, 199, 201, 203, 209, 211, 223, 225, 230, 231

Offset

1,3

Comments

Here "anti-palindromic" means the expansion is of the form x.y, where the complement of y is the reverse of x (allowing leading or trailing zeros). Here we do not insist that the base-phi representation be "canonical" (that is, we do not insist that xy contains no 11).

Links

Table of n, a(n) for n=1..61.

George Bergman, A number system with an irrational base, Math. Mag. 31 (1957), pp. 98-110.

Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023.

Formula

There is a 193-state automaton accepting the Zeckendorf representation of the members of this sequence.

Example

For example, one base-phi representation of 13 is 00100001.01111011.

Crossrefs

Cf. A105424, A341722.

Sequence in context: A023367 A291880 A047426 * A323527 A316091 A026487

Adjacent sequences: A362778 A362779 A362780 * A362782 A362783 A362784

Keyword

nonn,base

Author

Jeffrey Shallit, May 03 2023

Status

approved