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A059389 Sums of two nonzero Fibonacci numbers.

%I A059389 #35 Feb 16 2015 04:11:30
%S A059389 2,3,4,5,6,7,8,9,10,11,13,14,15,16,18,21,22,23,24,26,29,34,35,36,37,
%T A059389 39,42,47,55,56,57,58,60,63,68,76,89,90,91,92,94,97,102,110,123,144,
%U A059389 145,146,147,149,152,157,165,178,199,233,234,235,236,238,241,246,254,267
%N A059389 Sums of two nonzero Fibonacci numbers.
%C A059389 The sums of two distinct nonzero Fibonacci numbers is essentially the same sequence: 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 18, 21, ... (only 2 is missing), since F(i) + F(i) = F(i-2) + F(i+1). - _Colm Mulcahy_, Mar 02 2008
%C A059389 To elaborate on Mulcahy's comment above: all terms of A078642 are in this sequence; those are numbers with two distinct representations as the sum of two Fibonacci numbers, which are, as Alekseyev proved, numbers of the form 2*F(i) greater than 2. - _Alonso del Arte_, Jul 07 2013
%H A059389 T. D. Noe, <a href="/A059389/b059389.txt">Table of n, a(n) for n = 1..1000</a>
%F A059389 a(1) = 2 and for n >= 2 a(n) = F_(trinv(n-2)+2) + F_(n-((trinv(n-2)*(trinv(n-2)-1))/2)) where F_n is the n-th Fibonacci number, F_1 = 1 F_2 = 1 F_3 = 2 ... and the definition of trinv(n) is in A002262. - Noam Katz (noamkj(AT)hotmail.com), Feb 04 2001
%F A059389 log a(n) ~ sqrt(n log phi) where phi is the golden ratio A001622. There are (log x/log phi)^2 + O(log x) members of this sequence up to x. - _Charles R Greathouse IV_, Jul 24 2012
%e A059389 10 is in the sequence because 10 = 2 + 8.
%e A059389 11 is in the sequence because 11 = 3 + 8.
%e A059389 12 is not in the sequence because no pair of Fibonacci numbers adds up to 12.
%p A059389 N:= 1000: # to get all terms <= N
%p A059389 R:= NULL:
%p A059389 for j from 1 do
%p A059389   r:= combinat:-fibonacci(j);
%p A059389   if r > N then break fi;
%p A059389   R:= R, r;
%p A059389 end:
%p A059389 R:= {R}:
%p A059389 select(`<=`, {seq(seq(r+s, s=R),r=R)},N);
%p A059389 # if using Maple 11 or earlier, uncomment the next line
%p A059389 # sort(convert(%,list)); # _Robert Israel_, Feb 15 2015
%t A059389 max = 13; Select[Union[Total/@Tuples[Fibonacci[Range[2, max]], {2}]], # <= Fibonacci[max] &] (* _Harvey P. Dale_, Mar 13 2011 *)
%o A059389 (PARI) list(lim)=my(upper=log(lim*sqrt(5))\log((1+sqrt(5))/2)+1, t, tt, v=List([2])); if(fibonacci(t)>lim,t--); for(i=3,upper, t=fibonacci(i); for(j=2,i-1,tt=t+fibonacci(j); if(tt>lim, break, listput(v,tt)))); vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 24 2012
%Y A059389 Cf. A000045, A059390 (complement). Similar in nature to A048645. Essentially the same as A084176. Intersection with A049997 is A226857.
%K A059389 nonn,easy
%O A059389 1,1
%A A059389 Avi Peretz (njk(AT)netvision.net.il), Jan 29 2001
%E A059389 More terms from Larry Reeves (larryr(AT)acm.org), Jan 31 2001