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%I #45 Jul 15 2023 06:38:53
%S 1,1,2,3,3,5,5,5,8,8,8,5,10,13,12,12,13,10,7,15,18,21,16,20,20,16,21,
%T 18,15,7,17,25,27,27,34,29,20,32,32,32,20,29,34,27,27,25,17,9,24,32,
%U 40,33,45,45,39,55,50,45,24,40,52,48,48,52,40,24,45,50,55,39,45,45
%N Number of ways not ending in 011 to write n in base phi.
%C Old name was: Number of ways to write n in base phi.
%C phi = (1+sqrt(5))/2. Base phi is also called golden ratio base or phinary. In base phi, we can replace two consecutive 1's with a one in the column to the left; e.g., "011" = "100".
%C Conjecture: a(A005248(k)) = 2k+1 for k=1,2,...(cf. Theorem 2 in the paper by Carlitz.) - _Michel Dekking_, Nov 14 2021
%C This conjecture is proved in the paper "Counting base phi representations". - _Michel Dekking_, Jul 15 2023
%H L. Carlitz, <a href="https://fq.math.ca/Scanned/6-4/carlitz.pdf">Fibonacci Representations</a>, Fibonacci Quarterly, volume 6, number 4, October 1968, pages 193-220.
%H Michel Dekking and Ad van Loon, <a href="https://arxiv.org/abs/2304.11387">Counting base phi representations</a>, arXiv:2304.11387 [math.NT], 2023.
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/phigits.html">Base phi calculator</a>.
%e a(3) = 3, because 3 in base phi = 10.1111 = 11.01 = 100.01.
%Y Cf. A001622, A130600, A130601.
%K nonn,base
%O 0,3
%A Gilian Breysens, Jul 11 2017
%E Name corrected by _Michel Dekking_, Sep 09 2021
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