A078642 Numbers with two representations as the sum of two Fibonacci numbers.

4, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338, 126491972

Offset: 1

Author

Joseph L. Pe, Dec 12 2002

Keywords

• nonn
• easy

Comments

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, Dec 20 2002. Proved by Max Alekseyev, Mar 02 2007.

From Max Alekseyev, 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)

Examples

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

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (1,1)

Crossrefs

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.

Programs

Mathematica

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 *)

PARI

a(n)=([0,1; 1,1]^(n-1)*[4;6])[1,1] \\ Charles R Greathouse IV, Oct 07 2015

PARI

Vec(2*x*(2 + x) / (1 - x - x^2) + O(x^60)) \\ Colin Barker, Jan 29 2017

Formula

a(n) = 2F(n + 2), where F(n) is the n-th 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). - Philippe Deléham, Nov 19 2008

a(n) = 2F(n + 2) = F(n) + F(n + 3), where F(1) = F(2) = 1. - Alonso del Arte, Jul 07 2013

a(n) = (2^(-n)*((1-r)^n*(-3+r) + (1+r)^n*(3+r))) / r where r=sqrt(5). - Colin Barker, Jan 29 2017