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%I #64 Feb 13 2021 01:10:39
%S 0,1,1,2,2,12,12,112,112,1112,1112,11112,11112,111112,111112,1111112,
%T 1111112,11111112,11111112,111111112,111111112,1111111112,1111111112,
%U 11111111112,11111111112,111111111112,111111111112,1111111111112,1111111111112,11111111111112,11111111111112
%N 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.
%C In base 10, the corresponding sequence is A069638 and is periodic.
%H Colin Barker, <a href="/A305753/b305753.txt">Table of n, a(n) for n = 0..1000</a>
%H B. Chen, R. Chen, J. Guo, S. Lee et al., <a href="http://arxiv.org/abs/1808.04304">On Base 3/2 and its Sequences</a>, arXiv:1808.04304 [math.NT], 2018.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,10,-10).
%F From _Colin Barker_, Jun 14 2018: (Start)
%F Generating function: x*(1 - 3*x)*(1 + 3*x) / ((1 - x)*(1 - 10*x^2)).
%F a(n) = (10^(n/2) + 80) / 90 for n>0.
%F a(n) = (10^((n-1)/2) + 8) / 9 for n>0.
%F a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) for n>4.
%F (End)
%e Write decimal numbers as x_10, base-3/2 numbers as x_b (see A024629).
%e We have a(1) = 1, a(2) = 2 (in both bases).
%e Adding, we get 1+2 = 3_10 = 20_b, and sorting the digits gives a(3) = 2_b = 2_10.
%e Adding 2 and 2 we get 4_10 = 21_b, and sorting the digits gives a(4) = 12_b = (7/2)_10.
%e Adding 2 and 7/2 we get (11/2)_10 = 201_b, and sorting the digits gives a(5) = 12_b = (7/2)_10.
%e 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.
%e 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.
%e And so on.
%o (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
%Y Cf. A000045, A024629, A069638, A237575, A305880.
%Y This is A047855 with terms repeated. - _N. J. A. Sloane_, Jun 19 2018
%K nonn,base
%O 0,4
%A _Tanya Khovanova_ and PRIMES STEP Senior group, Jun 09 2018
%E Edited by _N. J. A. Sloane_, Jun 22 2018
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