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Non-Fibonacci numbers that are both a sum and a difference of two Fibonacci numbers.
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%I #24 Oct 31 2023 17:36:57

%S 4,6,7,10,11,16,18,26,29,42,47,68,76,110,123,178,199,288,322,466,521,

%T 754,843,1220,1364,1974,2207,3194,3571,5168,5778,8362,9349,13530,

%U 15127,21892,24476,35422,39603,57314,64079,92736,103682,150050,167761,242786

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

%C Intersection of A001690 and A007298 and A084176.

%C 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.

%C For relation with Lucas numbers, see formula section.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,1).

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

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

%F G.f.: -x*(4+6*x+3*x^2+4*x^3)/(-1+x^2+x^4) . - _R. J. Mathar_, Jan 13 2023

%F a(n) = a(n-2) + a(n-4) for n > 4. - _Christian Krause_, Oct 31 2023

%e 6 is a term because 6 = Fibonacci(1) + Fibonacci(5) = Fibonacci(6) - Fibonacci(3).

%e 16 is a term because 16 = Fibonacci(6) + Fibonacci(6) = Fibonacci(8) - Fibonacci(5).

%e 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.

%t 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 *)

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

%K nonn,easy

%O 1,1

%A _Altug Alkan_, May 04 2016