

A059389


Sums of two nonzero Fibonacci numbers.


7



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, 39, 42, 47, 55, 56, 57, 58, 60, 63, 68, 76, 89, 90, 91, 92, 94, 97, 102, 110, 123, 144, 145, 146, 147, 149, 152, 157, 165, 178, 199, 233, 234, 235, 236, 238, 241, 246, 254, 267
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OFFSET

1,1


COMMENTS

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(i2) + F(i+1).  Colm Mulcahy, Mar 02 2008
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


LINKS



FORMULA

a(1) = 2 and for n >= 2 a(n) = F_(trinv(n2)+2) + F_(n((trinv(n2)*(trinv(n2)1))/2)) where F_n is the nth 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
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


EXAMPLE

10 is in the sequence because 10 = 2 + 8.
11 is in the sequence because 11 = 3 + 8.
12 is not in the sequence because no pair of Fibonacci numbers adds up to 12.


MAPLE

N:= 1000: # to get all terms <= N
R:= NULL:
for j from 1 do
r:= combinat:fibonacci(j);
if r > N then break fi;
R:= R, r;
end:
R:= {R}:
select(`<=`, {seq(seq(r+s, s=R), r=R)}, N);
# if using Maple 11 or earlier, uncomment the next line


MATHEMATICA

max = 13; Select[Union[Total/@Tuples[Fibonacci[Range[2, max]], {2}]], # <= Fibonacci[max] &] (* Harvey P. Dale, Mar 13 2011 *)


PROG

(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, i1, tt=t+fibonacci(j); if(tt>lim, break, listput(v, tt)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2012


CROSSREFS



KEYWORD

nonn,easy


AUTHOR

Avi Peretz (njk(AT)netvision.net.il), Jan 29 2001


EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Jan 31 2001


STATUS

approved



