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

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, 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, 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

Offset

1,2

Comments

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.

In other words we give preference to the smallest Fibonacci numbers.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..8253 (rows for n = 1..985 flattened)

Rémy Sigrist, PARI program

W. Steiner, The joint distribution of greedy and lazy Fibonacci expansions, Fib. Q., 43 (No. 1, 2005), 60-69.

Example

Triangle begins:
1 meaning 1 = 1
2 meaning 2 = 2
1 2 meaning 3 = 1+2
1 3 meaning 4 = 1+3
2 3 meaning 5 = 2+3
1 2 3 meaning 6 = 1+2+3 (and not the Zeckendorf expansion 1+5)
2 5 meaning 7 = 2+5

Maple

A112309 := proc(n)
    local z, d ;
    z := convert(A104326(n), base, 10) ;
    for d from 1 to nops(z) do
        if op(d, z) > 0 then
            printf("%d, ", combinat[fibonacci](d+1)) ;
        end if;
    end do:
end proc:
for n from 1 to 20 do
    A112309(n) ;
end do: # R. J. Mathar, Aug 28 2025

Mathematica

DeleteCases[IntegerDigits[Range[200], 2], {___, 0, 0, ___}]
A112309 = Map[DeleteCases[Reverse[#] Fibonacci[Range[Length[#]] + 1], 0] &, DeleteCases[IntegerDigits[-1 + Range[200], 2], {___, 0, 0, ___}]]
A112310 = Map[Length, A112309]
(* Peter J. C. Moses, Mar 03 2015 *)

Prog

(PARI) See Links section.

Crossrefs

Cf. A000045, A112310, A035517, A007895.

Sequence in context: A259177 A304036 A173442 * A160006 A060682 A352897

Adjacent sequences: A112306 A112307 A112308 * A112310 A112311 A112312

Keyword

nonn,tabf,easy

Author

N. J. A. Sloane, Dec 01 2005

Extensions

Extended by Ray Chandler, Dec 01 2005

Status

approved