A190182 Decimal expansion of (1+x+sqrt(8+2x))/4, where x=sqrt(15).

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

Offset: 1

Author

Clark Kimberling, May 05 2011

Keywords

• nonn
• cons
• easy

Comments

The rectangle R whose shape (i.e., length/width) is (1+x+sqrt(8+2x))/4, where x=sqrt(15), can be partitioned into golden rectangles and squares in a manner that matches the periodic continued fraction [r,1,1,r,1,1,...]. It can also be partitioned into squares so as to match the nonperiodic continued fraction [2,4,1,3,10,...] at A190183. For details, see A188635.

Examples

			2.210275532819020968778971352504887053304...
		

Links

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

Crossrefs

Cf. A188635, A190183, A189970.

Programs

Magma

[(1 + Sqrt(15) + Sqrt(8 + 2*Sqrt(15)))/4]; // G. C. Greubel, Dec 28 2017

Mathematica

r = (1 + 5^(1/2))/2;
FromContinuedFraction[{r, 1, 1, {r, 1, 1}}]
FullSimplify[%]
ContinuedFraction[%, 100] (*A190183*)
RealDigits[N[%%, 120]]    (*A190182*)
N[%%%, 40]
RealDigits[(1 + Sqrt[15] + Sqrt[8 + 2*Sqrt[15]])/4, 10, 100][[1]] (* G. C. Greubel, Dec 28 2017 *)

PARI

(1 + sqrt(15) + sqrt(8 + 2*sqrt(15)))/4 \\ G. C. Greubel, Dec 28 2017