1,1

A positive integer n has exactly two representations as the sum of two Fibonacci numbers if and only if n is twice a Fibonacci number and n>=4. Conjectured by John W. Layman (layman(AT)math.vt.edu), Dec 20 2002. Proved by Max Alekseyev, Mar 02 2007.

Comments from Max Alekseyev (maxale(AT)gmail.com), Mar 02 2007: (Start)

Suppose that the number m has exactly two representations as the sum of two Fibonacci numbers. There are three types of representations possible:

(I) the sum of two equal Fibonacci numbers

(II) the sum of two consecutive Fibonacci numbers

(III) the sum of two distinct non-consecutive Fibonacci numbers

Lemma. The two representations of m>2 must be of different types.

Proof. Two representations of m>2 both of type (I) are not possible as 2*F_n is a strictly increasing function for n>=2. Similarly, two representations of m both of type (II) are not possible as F_n + F_{n+1} is a strictly increasing function for n>=0. Finally, two representations of m both of type (III) are not possible as that would violate the property of the Fibonacci numeral system (the uniqueness of representation of all nonnegative integers).

Consider all possible pairs of representation types:

(I) and (II) are possible only for m=2: 2 = 2*F_1 = 2*F_2 = F_1+F_2 but m=2 has more than two different representations.

(II) and (III) are not possible together as that would again violate the property of the Fibonacci numeral system.

Finally, (I) and (III) gives rise to the sequence of a(n) = 2 * F_n = F_{n+1} + F_{n-1}. QED (End)

Index entries for sequences related to linear recurrences with constant coefficients

Tanya Khovanova, Recursive Sequences

a(n) = 2F(n-2), where F(n) is a Fibonacci number.

a(n)=a(n-1)+a(n-2), n>2 ; a(1)=4, a(2)=6 . G.f.: 2x*(2+x)/(1-x-x^2). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 19 2008]

16 has exactly two representations as the sum of Fibonacci numbers: 16 = 3 + 13 and 16 = 8 + 8. Hence 16 belongs to the sequence.

t = Split@ Sort@ Flatten@ Table[Fibonacci[i] + Fibonacci[j], {i, 2, 39}, {j, 2, i}]; Take[ t[[ # ]][[1]] & /@ Select[ Range@Length@t, Length[ t[[ # ]]] > 1 &], 36] (* Robert G. Wilson v *)

Essentially the same as A006355 = number of binary vectors of length n containing no singletons; and as A055389: a(0)=1, then twice the Fibonacci sequence.

Sequence in context: A027689 A023501 A050887 this_sequence A032416 A117149 A165186

Adjacent sequences: A078639 A078640 A078641 this_sequence A078643 A078644 A078645

nonn

Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Dec 12 2002

More terms from John W. Layman (layman(AT)math.vt.edu), Dec 20 2002

Further terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 14 2005