A362917 The part of n to the left of the decimal point in the Dekking-van-Loon-canonical base phi representation of n.

0, 1, 10, 11, 101, 1000, 1010, 1011, 10001, 10010, 10011, 10101, 100000, 100010, 100011, 100101, 101000, 101010, 101011, 1000001, 1000010, 1000011, 1000101, 1001000, 1001010, 1001011, 1010001, 1010010, 1010011, 1010101, 10000000

Offset

0,3

Comments

The part to the right of the decimal point, reversed, is given by A341722, that is, it is the same as in the Bergman-canonical representation. I asked Jeffrey Shallit to confirm this, and he provided the following verification using the Walnut Theorem-Prover:

[Walnut]$ eval sloane "?msd_fib An,x1,x2,y1,y2 ($saka(n,x1,y1) & $dvl(n,x2,y2)) => $equal(y1,y2)":

(saka(n,x1,y1))&dvl(n,x2,y2))):54 states - 66ms

((saka(n,x1,y1))&dvl(n,x2,y2)))=>equal(y1,y2))):2 states - 25ms

(A n , x1 , x2 , y1 , y2 ((saka(n,x1,y1))&dvl(n,x2,y2)))=>equal(y1,y2)))):1 states - 81ms

Total computation time: 264ms.

____

TRUE

References

Dekking, Michel, and Ad van Loon. "On the representation of the natural numbers by powers of the golden mean." arXiv preprint arXiv:2111.07544 (2021); Fib. Quart. 61:2 (May 2023), 105-118.

Links

Table of n, a(n) for n=0..30.

Michel Dekking and Ad van Loon, On the representation of the natural numbers by powers of the golden mean, arXiv:2111.07544 [math.NT], 15 Nov 2021.

Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023. (see chapter 10, page 28)

Example

The canonical base phi representations of the numbers 0 through 12 are:
0 = 0.
1 = 1.
2 = 10.01
3 = 11.01
4 = 101.01
5 = 1000.1001
6 = 1010.0001
7 = 1011.0001
8 = 10001.0001
9 = 10010.0101
10 = 10011.0101
11 = 10101.0101
12 = 100000.101001

Crossrefs

Cf. A341722, A362918.

Differs from A105424 at positions given by A003231.

Sequence in context: A303593 A124387 A171796 * A305379 A004685 A361335

Adjacent sequences: A362914 A362915 A362916 * A362918 A362919 A362920

Keyword

nonn,base,more

Author

N. J. A. Sloane, May 26 2023

Extensions

a(13)-a(32) from Hugo Pfoertner, May 26 2023

Status

approved