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

1, 1001, 10001, 10101, 10001001, 10100001, 100000001, 100010001, 100100101, 101000101, 101010101, 100000101001, 100010001001, 100100001001, 100101001001, 101000100001, 101010000001, 1000000000001, 1000001000001, 1000010010001, 1000100010001, 1000101010001, 1001000100101

Offset

1,2

Comments

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

Links

Casey Mongoven and T. D. Noe, Table of n, a(n) for n = 1..1000

Ron Knott, Integers written in base phi

Casey Mongoven, Music based on this sequence

Formula

a(n) = A014417(A269725(n))//01, where // denotes concatenation and the leading 0's are removed from a(1). - Peter Munn, Sep 10 2025

Example

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.

Mathematica

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 *)

Prog

(PARI)
a(n) = {
    my(p = 1, s=0);
    while(p <= n, p *= quadgen(5));
    while(n > 0,
        p /= quadgen(5);
        s *= 10;
        if(p <= n, n -= p; s++)
    );
s} \\ Chittaranjan Pardeshi, Dec 14 2025
(Python)
from sympy import S
def a(n):
    p = 1
    while p <= n:
        p *= S.GoldenRatio
    d = 0
    while n > 0:
        d *= 10
        p /= S.GoldenRatio
        if p <= n:
            n -= p
            d += 1
    return d
alist = [a(i) for i in range(1, 20)]
print(alist) # Chittaranjan Pardeshi, Dec 15 2025

Crossrefs

Cf. A001622, A014417, A055778, A105424, A130601, A269725, A341722.

Sequence in context: A259080 A100709 A372419 * A328947 A114387 A357774

Adjacent sequences: A130597 A130598 A130599 * A130601 A130602 A130603

Keyword

nonn,base

Author

Casey Mongoven, Aug 06 2007

Status

approved