A190259 -id:A190259 - cp's OEIS frontend

A190258 Decimal expansion of (x + sqrt(2 + 4x))/2, where x=sqrt(2).

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

Offset: 1

Clark Kimberling, May 06 2011

The rectangle R whose shape (i.e., length/width) is (x+sqrt(2+4x))/2, where x=sqrt(2), can be partitioned into rectangles of shapes sqrt(2) and 1 in a manner that matches the periodic continued fraction [x, 1, x, 1, ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [2,11,32,1,4,10,2,1,...] at A190259. For details, see A188635.

			2.090657850852244775710089635005221328095...

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

Cf. A190259, A190260, A188635.

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

r=2^(1/2);
FromContinuedFraction[{r,1, {r,1}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190258 *)
RealDigits[N[%%, 120]]     (* A190259 *)
N[%%%, 40]
RealDigits[(Sqrt[2]+Sqrt[2+4Sqrt[2]])/2,10,120][[1]] (* Harvey P. Dale, Jun 20 2021 *)

PARI
```
sqrt(1/2)+sqrt(1/2+sqrt(2))
```

A190261 Continued fraction of (1 + sqrt(1 + 2x))/2, where x=sqrt(2).

Original entry on oeis.org

1, 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, 8, 2, 1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 8, 2

Offset: 1

Clark Kimberling, May 06 2011

1 followed by A190259.

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

Cf. A190260, A190259.

ContinuedFraction((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]
ContinuedFraction[(1+Sqrt[1+2Sqrt[2]])/2,100] (* Harvey P. Dale, Jan 27 2013 *)

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

cp's OEIS Frontend

A190258 Decimal expansion of (x + sqrt(2 + 4x))/2, where x=sqrt(2).

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