A362970 Number of different "integer parts" of (possibly non-canonical) base-phi representations of n.

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

Offset

0,2

Comments

Natural numbers can have infinitely many non-canonical expansions in base phi using the digits {0,1} only. For example, 2 = 10.01 = 10.0011 = 10.001011 = ... and so forth. However, there will only be finitely many possible distinct integer parts (the part to the left of the decimal point). a(n) is then the number of possibilities.

Links

Table of n, a(n) for n=0..79.

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

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

Formula

There is a linear representation of rank 28 to compute a(n).

Example

For n = 14 the a(14) = 6 possible integer parts are 11010, 11011, 11100, 100010, 100011, 100100.

Crossrefs

Sequence in context: A241092 A055656 A078571 * A194343 A071860 A358472

Adjacent sequences: A362967 A362968 A362969 * A362971 A362972 A362973

Keyword

nonn,base

Author

Jeffrey Shallit, May 11 2023

Status

approved