A272632 - OEIS

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

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OFFSET

1,1

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.

LINKS

Table of n, a(n) for n=1..46.

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

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

Sequence in context: A268574 A023631 A080641 * A229744 A191920 A379239

Adjacent sequences: A272629 A272630 A272631 * A272633 A272634 A272635

KEYWORD

nonn,easy

AUTHOR

Altug Alkan, May 04 2016

STATUS

approved