A305753 A base-3/2 sorted Fibonacci sequence that starts with a(0) = 0 and a(1) = 1. The terms are interpreted as numbers written in base 3/2. To get a(n+2), add a(n) and a(n+1), write the result in base 3/2 and sort the "digits" into increasing order, omitting all zeros.

0, 1, 1, 2, 2, 12, 12, 112, 112, 1112, 1112, 11112, 11112, 111112, 111112, 1111112, 1111112, 11111112, 11111112, 111111112, 111111112, 1111111112, 1111111112, 11111111112, 11111111112, 111111111112, 111111111112, 1111111111112, 1111111111112, 11111111111112, 11111111111112

Offset

0,4

Comments

In base 10, the corresponding sequence is A069638 and is periodic.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

B. Chen, R. Chen, J. Guo, S. Lee et al., On Base 3/2 and its Sequences, arXiv:1808.04304 [math.NT], 2018.

Index entries for linear recurrences with constant coefficients, signature (1,10,-10).

Formula

From Colin Barker, Jun 14 2018: (Start)

Generating function: x*(1 - 3*x)*(1 + 3*x) / ((1 - x)*(1 - 10*x^2)).

a(n) = (10^(n/2) + 80) / 90 for n>0.

a(n) = (10^((n-1)/2) + 8) / 9 for n>0.

a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) for n>4.

(End)

Example

Write decimal numbers as x_10, base-3/2 numbers as x_b (see A024629).
We have a(1) = 1, a(2) = 2 (in both bases).
Adding, we get 1+2 = 3_10 = 20_b, and sorting the digits gives a(3) = 2_b = 2_10.
Adding 2 and 2 we get 4_10 = 21_b, and sorting the digits gives a(4) = 12_b = (7/2)_10.
Adding 2 and 7/2 we get (11/2)_10 = 201_b, and sorting the digits gives a(5) = 12_b = (7/2)_10.
Adding (7/2)_10 and (7/2)_10 we get 7_10 = 211_b, and sorting the digits gives a(6) = 112_b = (23/4)_10.
Adding (7/2)_10 and (23/4)_10 we get (37/4)_10 = 2011_b, and sorting the digits gives a(7) = 112_b = (23/4)_10.
And so on.

Prog

(PARI) concat(0, Vec(x*(1 - 3*x)*(1 + 3*x) / ((1 - x)*(1 - 10*x^2)) + O(x^40))) \\ Colin Barker, Jun 19 2018

Crossrefs

Cf. A000045, A024629, A069638, A237575, A305880.

This is A047855 with terms repeated. - N. J. A. Sloane, Jun 19 2018

Sequence in context: A341432 A092144 A224497 * A181813 A059187 A350329

Adjacent sequences: A305750 A305751 A305752 * A305754 A305755 A305756

Keyword

nonn,base

Author

Tanya Khovanova and PRIMES STEP Senior group, Jun 09 2018

Extensions

Edited by N. J. A. Sloane, Jun 22 2018

Status

approved