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A112309 Triangle read by rows: row n gives terms in lazy Fibonacci representation of n. 3

%I

%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 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 *)

%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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Last modified July 11 23:53 EDT 2020. Contains 335654 sequences. (Running on oeis4.)