A176398 Decimal expansion of 3+sqrt(10).

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

Offset: 1

Author

Klaus Brockhaus, Apr 16 2010

Keywords

• cons
• nonn
• easy

Comments

Continued fraction expansion of 3+sqrt(10) is A010722.

This is the shape of a 6-extension rectangle; see A188640 for definitions. - Clark Kimberling, Apr 09 2011

c^n = c*A005668(n) + A005668(n-1). - Gary W. Adamson, Apr 04 2024

Examples

			6.16227766016837933199...
		

Links

• Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
• Wikipedia, Metallic mean.
• Index entries for algebraic numbers, degree 2

Crossrefs

Cf. A010467 (decimal expansion of sqrt(10)), A010722 (all 6's sequence).

Cf. A049310.

Programs

Maple

Digits:=100; evalf(3+sqrt(10)); # Wesley Ivan Hurt, Mar 07 2014

Mathematica

r=6; t=(r+(4+r^2)^(1/2))/2; RealDigits[N[t,130]][[1]] (* Clark Kimberling, Apr 09 2011 *)

PARI

3+sqrt(10) \\ Charles R Greathouse IV, Jul 24 2013

Formula

a(n) = A010467(n) for n >= 2.

Equals exp(arcsinh(3)), since arcsinh(x) = log(x+sqrt(x^2+1)). - Stanislav Sykora, Nov 01 2013

Equals lim_{n->oo} S(n, 2*sqrt(10))/ S(n-1, 2*sqrt(10)), with the S-Chebyshev polynomials (see A049310). - Wolfdieter Lang, Nov 15 2023