A189970 Decimal expansion of (1 + x + sqrt(14+10*x))/4, where x=sqrt(5).

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

Offset: 1

Author

Clark Kimberling, May 05 2011

Keywords

• nonn
• cons
• easy

Comments

Let R denote a rectangle whose shape (i.e., length/width) is (1 + x + sqrt(14+10*x))/4, where x=sqrt(5). This rectangle can be partitioned into golden rectangles and squares in a manner that matches the periodic continued fraction [r,1,r,1,r,1,r,1,...]. It can also be partitioned into squares so as to match the nonperiodic continued fraction [2,3,6,3,...] at A189971. For details, see A188635.

Decimal expansion of sqrt(r + r*sqrt(r + r*sqrt(r + ...))), where r = (1 + sqrt(5))/2 = golden ratio. - Ilya Gutkovskiy, Aug 24 2015

A quartic integer. - Charles R Greathouse IV, Aug 29 2015

Examples

			2.31651242917313233045161321161782337624579...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Index entries for algebraic numbers, degree 4

Crossrefs

Cf. A188635, A001622, A189971, A190157.

Programs

Magma

(1 + Sqrt(5) + Sqrt(14 + 10*Sqrt(5)) )/4; // G. C. Greubel, Jan 12 2018

Mathematica

r = (1 + 5^(1/2))/2;
FromContinuedFraction[{r, 1, {r, 1}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A189971 *)
RealDigits[N[%%, 120]]     (* A189970 *)
N[%%%, 40]
RealDigits[(1+Sqrt[5]+Sqrt[14+10Sqrt[5]])/4,10,120][[1]] (* Harvey P. Dale, Sep 24 2015 *)

PARI

default(realprecision,1000);x=sqrt(5);(1+x+sqrt(14+10*x))/4 \\ Anders Hellström, Aug 24 2015

PARI

polrootsreal(x^4-x^3-2*x^2-2*x-1)[2] \\ Charles R Greathouse IV, Aug 29 2015