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

%I A062877 #25 Jan 05 2025 19:51:36
%S A062877 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,
%T A062877 48,49,50,52,53,54,55,89,90,91,92,94,95,96,97,102,103,104,105,107,108,
%U A062877 109,110,123,124,125,126,128,129,130,131,136,137,138,139,141,142,143,144
%N A062877 0 and numbers representable as a sum of distinct odd-indexed Fibonacci numbers.
%H A062877 R. J. Mathar, <a href="/A062877/b062877.txt">Table of n, a(n) for n = 0..3071</a>
%H A062877 A. Karttunen, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/42-1/quartkarttunen01_2004.pdf">On Pascal's Triangle Modulo 2 in Fibonacci Representation</a>, The Fibonacci Quarterly, Vol. 42, #1 (2004) pp. 38-46.
%e A062877 F_1 = 1,
%e A062877 F_3 = 2,
%e A062877 F_1 + F_3 = 3,
%e A062877 F_5 = 5,
%e A062877 F_5 + F_1 = 6,
%e A062877 F_5 + F_3 = 7,
%e A062877 F_5 + F_3 + F_1 = 8,
%e A062877 F_7 = 13, ...
%p A062877 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));
%p A062877 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;
%p A062877 # alternative
%p A062877 isA062877 := proc(n)
%p A062877     local fset,fidx,ps ;
%p A062877     if n = 0 then
%p A062877         return true;
%p A062877     end if;
%p A062877     fset := {} ;
%p A062877     for fidx from 1 by 2 do
%p A062877         if combinat[fibonacci](fidx) >n then
%p A062877             break;
%p A062877         end if;
%p A062877         fset := fset union {combinat[fibonacci](fidx)} ;
%p A062877     end do:
%p A062877     for ps in combinat[powerset](fset) do
%p A062877         if n = add(fidx,fidx=ps) then
%p A062877             return true;
%p A062877         end if;
%p A062877     end do:
%p A062877     return false;
%p A062877 end proc: # _R. J. Mathar_, Aug 22 2016
%t A062877 Take[Union[Total/@Subsets[Fibonacci[Range[1,20,2]]]],70](* _Harvey P. Dale_, Dec 21 2013 *)
%o A062877 (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
%Y A062877 A062878 gives the positions of A050614(n) in this sequence. A062879 is bisection.
%Y A062877 A036796(n) - 1.
%Y A062877 Cf. A022290 (even-indexed Fibonaccis), A054204.
%K A062877 nonn
%O A062877 0,3
%A A062877 _Antti Karttunen_, Jun 26 2001