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

1,1

Comments

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.

Links

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

Index entries for algebraic numbers, degree 4.

Formula

Minimal polynomial: x^4 - 2*x^2 - 4*x - 2. - Amiram Eldar, Jun 01 2026

Example

2.090657850852244775710089635005221328095...

Mathematica

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

Prog

(PARI) sqrt(1/2)+sqrt(1/2+sqrt(2))
(Magma) [(Sqrt(2) + Sqrt(2+4*Sqrt(2)))/2]; // G. C. Greubel, Dec 26 2017

Crossrefs

Cf. A190259 (continued fraction), A190260, A188635.

Sequence in context: A105819 A153616 A389268 * A161119 A019750 A308473

Adjacent sequences: A190255 A190256 A190257 * A190259 A190260 A190261

Keyword

nonn,easy,cons,changed

Author

Clark Kimberling, May 06 2011

Status

approved