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A362970
Number of different "integer parts" of (possibly non-canonical) base-phi representations of n.
0
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
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
KEYWORD
nonn,base
AUTHOR
Jeffrey Shallit, May 11 2023
STATUS
approved