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A362716
Sum of the bits of the "integer part" of the base-phi representation of n.
2
0, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 3, 1, 2, 2, 3, 2, 3, 1, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 4, 1, 2, 2, 3, 2, 3, 2, 3, 3, 3, 4, 2, 3, 3, 4, 3, 4, 1, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 4, 2, 3, 3, 4, 3, 4, 2, 3, 3, 3, 4, 3, 4, 3, 4, 4, 4, 5, 1, 2, 2, 3, 2, 3, 2, 3, 3, 3
OFFSET
0,5
COMMENTS
The phi-representation of n is the (essentially) unique way to write n = Sum_{j=L..R} b(j)*phi^j, where b(j) is in {0,1} and -oo < L <= 0 <= R, where phi = (1+sqrt(5))/2, subject to the condition that b(j)b(j+1) != 1. The "integer" part is the string of bits b(R)b(R-1)...b(1)b(0).
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 19 for a(n).
EXAMPLE
For n = 20 we have n = phi^6 + phi^1 + phi^(-2) + phi^(-6), and the "integer part" has 2 terms, so a(20) = 2.
CROSSREFS
Cf. A055778.
Sequence in context: A025877 A219182 A184171 * A133989 A029398 A025817
KEYWORD
nonn
AUTHOR
Jeffrey Shallit, Apr 30 2023
STATUS
approved