OFFSET
1,1
COMMENTS
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.
LINKS
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).
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
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Altug Alkan, May 04 2016
STATUS
approved