A190258 -id:A190258 - cp's OEIS frontend

A190259 Continued fraction of (x + sqrt(2 + 4x))/2, where x=sqrt(2).

2, 11, 32, 1, 4, 10, 2, 1, 1, 3, 1, 1, 5, 2, 3, 2, 1, 4, 2, 3, 2, 41, 1, 2, 1, 1, 3, 4, 1, 35, 1, 5, 1, 29661, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 5, 2, 2, 2, 1, 1, 1, 5, 15, 2, 1, 1, 1, 2, 7, 1, 1, 1, 13, 1, 1, 1, 1, 20, 2, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 3, 14, 1

Offset: 1

Clark Kimberling, May 06 2011

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

Cf. A188635, A190258.

ContinuedFraction((Sqrt(2) + Sqrt(2+4*Sqrt(2)))/2); // G. C. Greubel, Dec 26 2017

(See A190258.)
ContinuedFraction[(Sqrt[2]+Sqrt[2+4Sqrt[2]])/2,100] (* Harvey P. Dale, Jun 16 2016 *)

contfrac((sqrt(2) + sqrt(2+4*sqrt(2)))/2) \\ G. C. Greubel, Dec 26 2017

Definition clarified by Harvey P. Dale, Jun 16 2016

A190260 Decimal expansion of (1 + sqrt(1 + 2*x))/2, where x=sqrt(2).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 06 2011

The rectangle R whose shape (i.e., length/width) is (1+sqrt(1+2x))/2, where x=sqrt(2), can be partitioned into rectangles of shapes 1 and sqrt(2) in a manner that matches the periodic continued fraction [1, x, 1, x, ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [1, 2,11,32,1,4,10,2,1,...] at A190261. For details, see A188635.

			1.478318343478515956422104436385022215253...

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

Cf. A188635, A190262, A190258.

[(1+Sqrt(1+2*Sqrt(2)))/2]; // G. C. Greubel, Dec 26 2017

r=2^(1/2);
FromContinuedFraction[{1, r, {1, r}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190261 *)
RealDigits[N[%%, 120]]     (* A190260 *)
N[%%%, 40]

(1+sqrt(1+2*sqrt(2)))/2 \\ G. C. Greubel, Dec 26 2017

cp's OEIS Frontend

A190259 Continued fraction of (x + sqrt(2 + 4x))/2, where x=sqrt(2).

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