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%I #16 Mar 23 2023 15:55:49
%S 1,2,1,2,1,3,2,3,1,2,3,2,5,1,2,5,1,3,5,2,3,5,1,2,3,5,1,3,8,2,3,8,1,2,
%T 3,8,2,5,8,1,2,5,8,1,3,5,8,2,3,5,8,1,2,3,5,8,2,5,13,1,2,5,13,1,3,5,13,
%U 2,3,5,13,1,2,3,5,13,1,3,8,13,2,3,8,13,1,2,3,8,13,2,5,8,13,1,2,5,8,13,1,3
%N Triangle read by rows: row n gives terms in lazy Fibonacci representation of n.
%C Write n as a sum c_2 F_2 + c_3 F_3 + ..., where the F_i are Fibonacci numbers and the c_i are 0 or 1. The lazy expansion is the minimal one in the lexicographic order, in contrast to the Zeckendorf expansion (A035517, A007895), which is the maximal one.
%C In other words we give preference to the smallest Fibonacci numbers.
%H Rémy Sigrist, <a href="/A112309/b112309.txt">Table of n, a(n) for n = 1..8253</a> (rows for n = 1..985 flattened)
%H Rémy Sigrist, <a href="/A112309/a112309.gp.txt">PARI program</a>
%H W. Steiner, <a href="http://www.fq.math.ca/Papers1/43-1/paper43-1-8.pdf">The joint distribution of greedy and lazy Fibonacci expansions</a>, Fib. Q., 43 (No. 1, 2005), 60-69.
%e Triangle begins:
%e 1 meaning 1 = 1
%e 2 meaning 2 = 2
%e 1 2 meaning 3 = 1+2
%e 1 3 meaning 4 = 1+3
%e 2 3 meaning 5 = 2+3
%e 1 2 3 meaning 6 = 1+2+3 (and not the Zeckendorf expansion 1+5)
%e 2 5 meaning 7 = 2+5
%t DeleteCases[IntegerDigits[Range[200], 2], {___, 0, 0, ___}]
%t A112309 = Map[DeleteCases[Reverse[#] Fibonacci[Range[Length[#]] + 1], 0] &, DeleteCases[IntegerDigits[-1 + Range[200], 2], {___, 0, 0, ___}]]
%t A112310 = Map[Length, A112309]
%t (* _Peter J. C. Moses_, Mar 03 2015 *)
%o (PARI) See Links section.
%Y Cf. A000045, A112310, A035517, A007895.
%K nonn,tabf,easy
%O 1,2
%A _N. J. A. Sloane_, Dec 01 2005
%E Extended by _Ray Chandler_, Dec 01 2005
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