A190177 Decimal expansion of (1+sqrt(2)+sqrt(7+6*sqrt(2)))/2.

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

Offset: 1

Author

Clark Kimberling, May 05 2011

Keywords

• nonn
• cons

Comments

Let R denote a rectangle whose shape (i.e., length/width) is (1+sqrt(2)+sqrt(7+6*sqrt(2)))/2. R can be partitioned into squares and silver rectangles in a manner that matches the periodic continued fraction [r,1,r,1,r,1,r,1,...], where r is the silver ratio: 1+sqrt(2)=[2,2,2,2,2,...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [3,5,1,2,1,1,1,2,...] at A190178. For details, see A188635.

Examples

			3.174673894034198922958074412217243642975...
		

Links

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

Crossrefs

Cf. A188635, A190178, A189970, A190179.

Programs

Magma

[(1+Sqrt(2)+Sqrt(7+6*Sqrt(2)))/2]; // G. C. Greubel, Dec 28 2017

Mathematica

r = 1 + 2^(1/2);
FromContinuedFraction[{r, 1, {r, 1}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190178 *)
RealDigits[N[%%, 120]]     (* A190177 *)
N[%%%, 40]
RealDigits[(1+Sqrt[2]+Sqrt[7+6*Sqrt[2]])/2, 10, 100][[1]] (* G. C. Greubel, Dec 28 2017 *)

PARI

(1+sqrt(2)+sqrt(7+6*sqrt(2)))/2 \\ G. C. Greubel, Dec 28 2017