A062877 0 and numbers representable as a sum of distinct odd-indexed Fibonacci numbers.

0, 1, 2, 3, 5, 6, 7, 8, 13, 14, 15, 16, 18, 19, 20, 21, 34, 35, 36, 37, 39, 40, 41, 42, 47, 48, 49, 50, 52, 53, 54, 55, 89, 90, 91, 92, 94, 95, 96, 97, 102, 103, 104, 105, 107, 108, 109, 110, 123, 124, 125, 126, 128, 129, 130, 131, 136, 137, 138, 139, 141, 142, 143, 144

Offset: 0

Author

Antti Karttunen, Jun 26 2001

Keywords

• nonn

Examples

			F_1 = 1,
F_3 = 2,
F_1 + F_3 = 3,
F_5 = 5,
F_5 + F_1 = 6,
F_5 + F_3 = 7,
F_5 + F_3 + F_1 = 8,
F_7 = 13, ...
		

Links

• R. J. Mathar, Table of n, a(n) for n = 0..3071
• A. Karttunen, On Pascal's Triangle Modulo 2 in Fibonacci Representation, The Fibonacci Quarterly, Vol. 42, #1 (2004) pp. 38-46.

Crossrefs

A062878 gives the positions of A050614(n) in this sequence. A062879 is bisection.

A036796(n) - 1.

Cf. A022290 (even-indexed Fibonaccis), A054204.

Programs

Maple

with(combinat); [seq(A062877(j),j=0..265)]; A062877 := n -> add((floor(n/(2^i)) mod 2)*fibonacci((2*i)+1),i=0..floor_log_2(n+1));
floor_log_2 := proc(n) local nn,i; nn := n; for i from -1 to n do if(0 = nn) then RETURN(i); fi; nn := floor(nn/2); od; end;
# alternative
isA062877 := proc(n)
    local fset,fidx,ps ;
    if n = 0 then
        return true;
    end if;
    fset := {} ;
    for fidx from 1 by 2 do
        if combinat[fibonacci](fidx) >n then
            break;
        end if;
        fset := fset union {combinat[fibonacci](fidx)} ;
    end do:
    for ps in combinat[powerset](fset) do
        if n = add(fidx,fidx=ps) then
            return true;
        end if;
    end do:
    return false;
end proc: # R. J. Mathar, Aug 22 2016

Mathematica

Take[Union[Total/@Subsets[Fibonacci[Range[1,20,2]]]],70](* Harvey P. Dale, Dec 21 2013 *)

PARI

my(m=Mod('x,'x^2-3*'x+1)); a(n) = subst(lift(subst(Pol(binary(n)), 'x,m)), 'x,2); \\ Kevin Ryde, Nov 25 2020