A118240 The part of n in base phi left of the decimal using a least-greedy algorithm representation.

0, 1, 1, 10, 11, 101, 111, 1010, 1011, 1101, 1110, 1111, 10101, 10111, 11010, 11011, 11101, 11111, 101010, 101011, 101101, 101110, 101111, 110101, 110111, 111010, 111011, 111101, 111110, 111111, 1010101, 1010111, 1011010, 1011011, 1011101

Offset

0,4

Comments

Uses least-greedy algorithm (start with largest possible power of phi, writing a 1 only when required, then work downward).

a(n) is also the left portion of the base-phi representation of n in Knott's representation which uses the least number of 0's, the most 1's, and in which the right-hand portion (see A362919) is finite. - N. J. A. Sloane, May 27 2023

Links

Table of n, a(n) for n=0..34.

Ron Knott, Phigits and the Base Phi representation.

Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023. [Note that this document has been revised multiple times.]

Example

6 = 111.01101010... in base phi using the least-greedy algorithm. The part to the left of the decimal is a(6) = 111.

Prog

constant (float): phi=(sqrt(5)+1)/2; variable (float): lphi=phi^floor[log(n)/log(phi)]; variable (float): rem=n; variable (integer): count=0; loop: while lphi>1 (count=count*10; lphi=lphi/phi; if(rem > lphi*phi) { rem=rem-lphi; count++; }}

Crossrefs

Cf. A055778, A104605, A118241, A105424, A362919, A362920.

Sequence in context: A368804 A037090 A171676 * A157845 A230297 A086084

Adjacent sequences: A118237 A118238 A118239 * A118241 A118242 A118243

Keyword

nonn,base

Author

Graeme McRae, Apr 17 2006

Extensions

a(1) corrected by N. J. A. Sloane, May 27 2023

Status

approved