A272632 Non-Fibonacci numbers that are both a sum and a difference of two Fibonacci numbers.

4, 6, 7, 10, 11, 16, 18, 26, 29, 42, 47, 68, 76, 110, 123, 178, 199, 288, 322, 466, 521, 754, 843, 1220, 1364, 1974, 2207, 3194, 3571, 5168, 5778, 8362, 9349, 13530, 15127, 21892, 24476, 35422, 39603, 57314, 64079, 92736, 103682, 150050, 167761, 242786

Offset: 1

Author

Altug Alkan, May 04 2016

Keywords

• nonn
• easy

Comments

Intersection of A001690 and A007298 and A084176.

Sequence focuses on the non-Fibonacci numbers because of the fact that all Fibonacci numbers are both the sum of two Fibonacci numbers and the difference of two Fibonacci numbers by definition of Fibonacci numbers.

For relation with Lucas numbers, see formula section.

Examples

			6 is a term because 6 = Fibonacci(1) + Fibonacci(5) = Fibonacci(6) - Fibonacci(3).
16 is a term because 16 = Fibonacci(6) + Fibonacci(6) = Fibonacci(8) - Fibonacci(5).
167761 is a term because it is not a Fibonacci number and 167761 = Fibonacci(24) + Fibonacci(26) = 46368 + 121393 and Fibonacci(24) + Fibonacci(26) = Fibonacci(27) - Fibonacci(23) by definition.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

Crossrefs

Cf. A000032, A055389, A001690, A007298, A084176.

Programs

Mathematica

mxf=30; {s,d} = Reap[Do[{a,b} = Fibonacci@{i,j}; Sow[a+b, 0]; Sow[a-b, 1], {i, mxf}, {j, i}]][[2]]; Complement[ Intersection[s, d], Fibonacci@ Range@ mxf] (* Giovanni Resta, May 04 2016 *)

Formula

a(2*n-1) = fibonacci(n+1) + fibonacci(n+3) =A000204(n+2) for n >= 1.

a(2*n) = 2*fibonacci(n+3) = A078642(n+1) for n >= 1.

G.f.: -x*(4+6*x+3*x^2+4*x^3)/(-1+x^2+x^4) . - R. J. Mathar, Jan 13 2023

a(n) = a(n-2) + a(n-4) for n > 4. - Christian Krause, Oct 31 2023