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

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

1,2

Comments

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.

Links

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

Index entries for algebraic numbers, degree 4.

Formula

Minimal polynomial: 2*x^4 - 4*x^3 + 2*x^2 - 1. - Amiram Eldar, Jun 01 2026

Example

1.478318343478515956422104436385022215253...

Mathematica

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

Prog

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

Crossrefs

Cf. A188635, A190261 (continued fraction), A190262, A190258.

Sequence in context: A275977 A151968 A115632 * A318383 A151958 A176778

Adjacent sequences: A190257 A190258 A190259 * A190261 A190262 A190263

Keyword

nonn,cons

Author

Clark Kimberling, May 06 2011

Status

approved