A190281 Decimal expansion of (1+sqrt(1+r))/r, where r=sqrt(2).

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

Offset: 1

Author

Clark Kimberling, May 07 2011

Keywords

• nonn
• cons

Comments

The rectangle R whose shape (i.e., length/width) is (1+sqrt(1+r))/r, where r=sqrt(2), can be partitioned into rectangles of shapes sqrt(2) and 2 in a manner that matches the periodic continued fraction [r, 2, r, 2, ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [1,1,4,6,1,2,2,2,1,1 ...] at A190282. For details, see A188635.

Examples

			1.805790894654357490440645557345527417829...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A190282, A190284.

Programs

Magma

(1 + Sqrt(1 + Sqrt(2)))/Sqrt(2); // G. C. Greubel, Jan 31 2018

Mathematica

r=2^(1/2)
FromContinuedFraction[{r,2, {r,2}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190282 *)
RealDigits[N[%%, 120]]     (* A190281 *)
N[%%%, 40]
RealDigits[(1 + Sqrt[1 + Sqrt[2]])/Sqrt[2], 10, 100][[1]] (* G. C. Greubel, Jan 31 2018 *)

PARI

(1 + sqrt(1 + sqrt(2)))/sqrt(2) \\ G. C. Greubel, Jan 31 2018