cp's OEIS Frontend

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

%I A362917 #29 May 27 2023 14:18:38
%S A362917 0,1,10,11,101,1000,1010,1011,10001,10010,10011,10101,100000,100010,
%T A362917 100011,100101,101000,101010,101011,1000001,1000010,1000011,1000101,
%U A362917 1001000,1001010,1001011,1010001,1010010,1010011,1010101,10000000
%N A362917 The part of n to the left of the decimal point in the Dekking-van-Loon-canonical base phi representation of n.
%C A362917 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:
%C A362917 [Walnut]$ eval sloane "?msd_fib An,x1,x2,y1,y2 ($saka(n,x1,y1) & $dvl(n,x2,y2)) => $equal(y1,y2)":
%C A362917 (saka(n,x1,y1))&dvl(n,x2,y2))):54 states - 66ms
%C A362917  ((saka(n,x1,y1))&dvl(n,x2,y2)))=>equal(y1,y2))):2 states - 25ms
%C A362917   (A n , x1 , x2 , y1 , y2 ((saka(n,x1,y1))&dvl(n,x2,y2)))=>equal(y1,y2)))):1 states - 81ms
%C A362917 Total computation time: 264ms.
%C A362917 ____
%C A362917 TRUE
%D A362917 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.
%H A362917 Michel Dekking and Ad van Loon, <a href="https://doi.org/10.48550/arXiv.2111.07544">On the representation of the natural numbers by powers of the golden mean</a>, arXiv:2111.07544 [math.NT], 15 Nov 2021.
%H A362917 Jeffrey Shallit, <a href="https://doi.org/10.48550/arXiv.2305.02672">Proving Properties of phi-Representations with the Walnut Theorem-Prover</a>, arXiv:2305.02672 [math.NT], 2023. (see chapter 10, page 28)
%e A362917 The canonical base phi representations of the numbers 0 through 12 are:
%e A362917 0 = 0.
%e A362917 1 = 1.
%e A362917 2 = 10.01
%e A362917 3 = 11.01
%e A362917 4 = 101.01
%e A362917 5 = 1000.1001
%e A362917 6 = 1010.0001
%e A362917 7 = 1011.0001
%e A362917 8 = 10001.0001
%e A362917 9 = 10010.0101
%e A362917 10 = 10011.0101
%e A362917 11 = 10101.0101
%e A362917 12 = 100000.101001
%Y A362917 Cf. A341722, A362918.
%Y A362917 Differs from A105424 at positions given by A003231.
%K A362917 nonn,base,more
%O A362917 0,3
%A A362917 _N. J. A. Sloane_, May 26 2023
%E A362917 a(13)-a(32) from _Hugo Pfoertner_, May 26 2023