A362920 -id:A362920 - cp's OEIS frontend

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

Graeme McRae, Apr 17 2006

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

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

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.]

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

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++;}}

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

A362919 a(n) is the right portion (reversed) 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 is finite.

Original entry on oeis.org

0, 0, 11, 1111, 1111, 1111, 1110, 111101, 111101, 111101, 111111, 111111, 111111, 111110, 111011, 111011, 111011, 111010, 11110101, 11110101, 11110101, 11110111, 11110111, 11110111, 11110110, 11111101, 11111101, 11111101, 11111111, 11111111, 11111111

Offset: 0

N. J. A. Sloane, May 27 2023

The left portion is given in A118240.

			The representations of the numbers 0 though 30 are:
  0 = 0.0
  1 = 1.0
  2 = 1.11
  3 = 10.1111
  4 = 11.1111
  5 = 101.1111
  6 = 111.0111
  7 = 1010.101111
  8 = 1011.101111
  9 = 1101.101111
  10 = 1110.111111
  11 = 1111.111111
  12 = 10101.111111
  13 = 10111.011111
  14 = 11010.110111
  15 = 11011.110111
  16 = 11101.110111
  17 = 11111.010111
  18 = 101010.10101111
  19 = 101011.10101111
  20 = 101101.10101111
  21 = 101110.11101111
  22 = 101111.11101111
  23 = 110101.11101111
  24 = 110111.01101111
  25 = 111010.10111111
  26 = 111011.10111111
  27 = 111101.10111111
  28 = 111110.11111111
  29 = 111111.11111111
  30 = 1010101.11111111

Ron Knott, Phigits and the Base Phi representation.
Ron Knott, Phigits and the Base Phi representation [Local copy, pdf only]
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.]

Cf. A118240, A362920.

cp's OEIS Frontend

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

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