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A111458 Numbers that cannot be represented as the sum of at most three Fibonacci numbers (with repetitions allowed).

%I A111458 #14 Jul 20 2023 07:21:04
%S A111458 33,46,51,53,54,67,72,74,75,80,82,83,85,86,87,88,101,106,108,109,114,
%T A111458 116,117,119,120,121,122,127,129,130,132,133,134,135,137,138,139,140,
%U A111458 141,142,143,156,161,163,164,169,171,172
%N A111458 Numbers that cannot be represented as the sum of at most three Fibonacci numbers (with repetitions allowed).
%H A111458 R. J. Mathar, <a href="/A111458/b111458.txt">Table of n, a(n) for n = 1..1000</a>
%e A111458 33 is neither a Fibonacci number nor can be written as the sum of two or three Fibonacci numbers.
%p A111458 isA111458 := proc(n) # returns true if n is in the sequence
%p A111458     local xi,yi,x,y,z ;
%p A111458     for xi from 0 do
%p A111458         x := A000045(xi) ;
%p A111458         if 3*x > n then
%p A111458             return true;
%p A111458         end if;
%p A111458         for yi from xi do
%p A111458             y := A000045(yi) ;
%p A111458             if x+2*y > n then
%p A111458                 break;
%p A111458             else
%p A111458                 z := n-x-y ;
%p A111458                 if isA000045(z) then # see isFib in A000045
%p A111458                     return false;
%p A111458                 end if;
%p A111458             end if;
%p A111458         end do:
%p A111458     end do:
%p A111458 end proc:
%p A111458 A111458 := proc(n)
%p A111458     option remember;
%p A111458     local a;
%p A111458     if n = 0 then
%p A111458         -1;
%p A111458     else
%p A111458         for a from procname(n-1)+1 do
%p A111458             if isA111458(a) then
%p A111458                 return a;
%p A111458             end if;
%p A111458         end do:
%p A111458     end if;
%p A111458 end proc: # _R. J. Mathar_, Sep 09 2015
%t A111458 FibQ[n_] := IntegerQ[Sqrt[5n^2+4]] || IntegerQ[Sqrt[5n^2-4]];
%t A111458 P[n_] := IntegerPartitions[n, 3, Select[Range[n], FibQ]];
%t A111458 Select[Range[1000], P[#] == {}&] (* _Jean-François Alcover_, Jul 20 2023 *)
%Y A111458 Cf. A000045, A000119, A179244, A135709.
%K A111458 nonn
%O A111458 1,1
%A A111458 _Stefan Steinerberger_, Nov 15 2005