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

Author

N. J. A. Sloane, Dec 01 2005

Keywords

• nonn
• tabf
• easy
• changed

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.

Examples

			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
		

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.

Crossrefs

Cf. A000045, A112310, A035517, A007895.

Programs

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 *)

PARI

See Links section.

Extensions

Extended by Ray Chandler, Dec 01 2005