A225667 Decimal expansion of 13-5*sqrt(5).

1, 8, 1, 9, 6, 6, 0, 1, 1, 2, 5, 0, 1, 0, 5, 1, 5, 1, 7, 9, 5, 4, 1, 3, 1, 6, 5, 6, 3, 4, 3, 6, 1, 8, 8, 2, 2, 7, 9, 6, 9, 0, 8, 2, 0, 1, 9, 4, 2, 3, 7, 1, 3, 7, 8, 6, 4, 5, 5, 1, 3, 7, 7, 2, 9, 4, 7, 3, 9, 5, 3, 7, 1, 8, 1, 0, 9, 7, 5, 5, 0, 2, 9, 2, 7, 9

Offset: 1

Author

Clark Kimberling, Jul 21 2013

Keywords

• nonn
• cons
• easy

Comments

Let d(n) = - 2*F(n) + h(2 + F(n+1), 1 + F(n+2)), where h = harmonic mean, F = A000045 (Fibonacci numbers). Then floor(d(n)) = 2F(n) + 1 for n>1, and limit(d(n)) = 13 - 5*sqrt(5).

Apart from leading digits the same as A132338, A109866, A094874 and A079585. - R. J. Mathar, Jul 30 2013

Examples

			13-5*sqrt(5) = 1.819660112501051517954131656343618822797...
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000045, A226765.

Programs

Mathematica

f[n_] := Fibonacci[n]; h[n_] := HarmonicMean[{2 + f[n + 1], 1 + f[n + 2]}]; x = Limit[-2 f[n] + h[n], n -> Infinity] (* "proof" *)
d = RealDigits[x, 10, 120][[1]] (* A225667 *)