A130601 -id:A130601 - cp's OEIS frontend

A133773 Number of runs (of equal bits) in the maximal "phinary" (A130601) representation of n.

1, 1, 3, 5, 3, 3, 7, 5, 5, 5, 9, 5, 5, 7, 5, 5, 5, 11, 9, 9, 7, 5, 7, 7, 9, 7, 7, 7, 13, 7, 7, 9, 7, 7, 7, 11, 9, 9, 7, 5, 7, 7, 9, 7, 7, 7, 15, 13, 13, 11, 9, 11, 11, 11, 9, 9, 7, 5, 9, 9, 11, 9, 9, 9, 13, 11, 11, 9, 7, 9, 9, 11, 9, 9, 9, 17, 9, 9, 11, 9, 9, 9, 13, 11, 11, 9, 7, 9, 9, 11, 9, 9, 9, 15, 13

Offset: 1

Casey Mongoven, Sep 23 2007

nonn

			A130601(3)=1101 because phi^1 + phi^0 + phi^-2 = 3; 1101 has 3 runs: 11,0,1. So a(3)=3.

Zeckendorf, E., Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège 41, 179-182, 1972.

Casey Mongoven, Table of n, a(n) for n = 1..199
Ron Knott, Using Powers of Phi to represent Integers.
Casey Mongoven, Music based on this sequence.

Cf. A133772, A130601.

A133776 Number of 0's in the maximal "phinary" (A130601) representation of n.

Original entry on oeis.org

0, 0, 1, 2, 1, 1, 3, 2, 2, 2, 4, 2, 2, 3, 2, 2, 2, 5, 4, 4, 3, 2, 3, 3, 4, 3, 3, 3, 6, 3, 3, 4, 3, 3, 3, 5, 4, 4, 3, 2, 3, 3, 4, 3, 3, 3, 7, 6, 6, 5, 4, 5, 5, 5, 4, 4, 3, 2, 4, 4, 5, 4, 4, 4, 6, 5, 5, 4, 3, 4, 4, 5, 4, 4, 4, 8, 4, 4, 5, 4, 4, 4, 6, 5, 5, 4, 3, 4, 4, 5, 4, 4, 4, 7, 6, 6, 5, 4, 5, 5, 5, 4, 4, 3, 2

Offset: 1

Casey Mongoven, Sep 23 2007

nonn

			A130601(7)=10101101, which has three 0's. So a(7)=3.

Zeckendorf, E., Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège 41, 179-182, 1972.

Casey Mongoven, Table of n, a(n) for n = 1..199
Ron Knott, Using Powers of Phi to represent Integers.

Cf. A133775, A130601.

A133774 Number of 1s in the maximal "phinary" (A130601) representation of n.

Original entry on oeis.org

1, 3, 3, 3, 6, 6, 5, 6, 6, 6, 5, 9, 9, 8, 9, 9, 9, 7, 8, 8, 9, 10, 9, 9, 8, 9, 9, 9, 7, 12, 12, 11, 12, 12, 12, 10, 11, 11, 12, 13, 12, 12, 11, 12, 12, 12, 9, 10, 10, 11, 12, 11, 11, 11, 12, 12, 13, 14, 12, 12, 11, 12, 12, 12, 10, 11, 11, 12, 13, 12, 12, 11, 12, 12, 12, 9, 15, 15, 14, 15, 15

Offset: 1

Casey Mongoven, Sep 23 2007

nonn

			A130601(4)=10101, which contains three 1s. Hence a(4)=3.

E. Zeckendorf, Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège 41, 179-182, 1972.

Casey Mongoven, Table of n, a(n) for n = 1..199
Ron Knott, Using Powers of Phi to represent Integers.

Cf. A055778, A130601.

A130600 Integers written in base phi, with the "decimal point" omitted.

Original entry on oeis.org

1, 1001, 10001, 10101, 10001001, 10100001, 100000001, 100010001, 100100101, 101000101, 101010101, 100000101001, 100010001001, 100100001001, 100101001001, 101000100001, 101010000001, 1000000000001, 1000001000001

Offset: 1

Casey Mongoven, Aug 06 2007

This is the "greedy" or "minimal" representation (see also A130601).

			If the decimal point were included, the sequence would read 1., 10.01, 100.01, 101.01, 1000.1001, 1010.0001, 10000.0001, 10001.0001, 10010.0101, 10100.0101, 10101.0101, ... Unfortunately these are not integers.
Examples: a(2)=1001 because phi^1+phi^-2 = 2, a(3) = 10001 because phi^2+phi^-2 = 3, a(4) = 10101 because phi^2+phi^0+phi^-2 = 4.

Casey Mongoven and T. D. Noe, Table of n, a(n) for n = 1..1000
Casey Mongoven, Music based on this sequence
Ron Knott, Integers written in base phi

Cf. A001622, A055778, A105424, A130601.

nn = 100; len = 2*Ceiling[Log[GoldenRatio, nn]]; Table[d = RealDigits[n, GoldenRatio, len]; last1 = Position[d[[1]], 1][[-1, 1]]; FromDigits[Take[d[[1]], last1]], {n, nn}] (* T. D. Noe, May 20 2011 *)

A289749 Number of ways not ending in 011 to write n in base phi.

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 5, 5, 8, 8, 8, 5, 10, 13, 12, 12, 13, 10, 7, 15, 18, 21, 16, 20, 20, 16, 21, 18, 15, 7, 17, 25, 27, 27, 34, 29, 20, 32, 32, 32, 20, 29, 34, 27, 27, 25, 17, 9, 24, 32, 40, 33, 45, 45, 39, 55, 50, 45, 24, 40, 52, 48, 48, 52, 40, 24, 45, 50, 55, 39, 45, 45

Offset: 0

Gilian Breysens, Jul 11 2017

Old name was: Number of ways to write n in base phi.
phi = (1+sqrt(5))/2. Base phi is also called golden ratio base or phinary. In base phi, we can replace two consecutive 1's with a one in the column to the left; e.g., "011" = "100".
Conjecture: a(A005248(k)) = 2k+1 for k=1,2,...(cf. Theorem 2 in the paper by Carlitz.) - Michel Dekking, Nov 14 2021
This conjecture is proved in the paper "Counting base phi representations". - Michel Dekking, Jul 15 2023

			a(3) = 3, because 3 in base phi = 10.1111 = 11.01 = 100.01.

L. Carlitz, Fibonacci Representations, Fibonacci Quarterly, volume 6, number 4, October 1968, pages 193-220.
Michel Dekking and Ad van Loon, Counting base phi representations, arXiv:2304.11387 [math.NT], 2023.
Ron Knott, Base phi calculator.

Cf. A001622, A130600, A130601.

Name corrected by Michel Dekking, Sep 09 2021

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A133773 Number of runs (of equal bits) in the maximal "phinary" (A130601) representation of n.

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A133776 Number of 0's in the maximal "phinary" (A130601) representation of n.

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A133774 Number of 1s in the maximal "phinary" (A130601) representation of n.

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A130600 Integers written in base phi, with the "decimal point" omitted.

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A289749 Number of ways not ending in 011 to write n in base phi.

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