A135558 Sums of three distinct nonzero Fibonacci numbers.

6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 73, 76, 77, 78, 79, 81, 84, 89, 90, 91, 92, 93, 94, 95

Offset: 1

Author

Colm Mulcahy, Feb 23 2008, Mar 02 2008

Keywords

• nonn

Comments

These numbers may have more than one such representation.

Links

• R. J. Mathar, Table of n, a(n) for n = 1..1000
• Colm Mulcahy, Additional Certainties, February 2008.

Crossrefs

Cf. A000045, A135709 (Complement).

Programs

Maple

isA135558 := proc(n) # returns true if n is in the sequence
    local xi,yi,x,y,z ;
    for xi from 2 do
        x := A000045(xi) ;
        if 3*x > n then
            return false;
        end if;
        for yi from xi+1 do
            y := A000045(yi) ;
            if x+2*y > n then
                break;
            else
                z := n-x-y ;
                if z >y and isA000045(z) then # see isFib in A000045
                    return true;
                end if;
            end if;
        end do:
    end do:
end proc:
A135558 := proc(n)
    option remember;
    local a;
    if n = 1 then
        6;
    else
        for a from procname(n-1)+1 do
            if isA135558(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Sep 09 2015

Mathematica

fibs[n_ /; n >= 6] := Reap[Module[{k = 1}, While[Fibonacci[k] < n, Sow[Fibonacci[k++]]]]][[2, 1]];
okQ[n_] := AnyTrue[IntegerPartitions[n, {3}, fibs[n]], Length[Union[#]] == 3&];
Select[Range[6, 100], okQ] (* Jean-François Alcover, Dec 12 2023 *)
Select[Total/@Subsets[Fibonacci[Range[2,12]],{3}]//Union,#<=100&] (* Harvey P. Dale, Jul 06 2025 *)

Extensions

More terms from N. J. A. Sloane, Mar 01 2008, corrected Mar 05 2008