A190289 Decimal expansion of (3+sqrt(21))/4.

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

Offset

1,2

Comments

The rectangle R whose shape (i.e., length/width) is (3+sqrt(21))/4, can be partitioned into rectangles of shapes 3/2 and 2 in a manner that matches the periodic continued fraction [3/2, 2, 3/2, 2, ...]. R can also be partitioned into squares so as to match the periodic continued fraction [1,1,8,1,1,2,1,1,8,1,1,2,,...]. For details, see A188635.

Links

Table of n, a(n) for n=1..120.

Index entries for algebraic numbers, degree 2.

Formula

Minimal polynomial: 4*x^2 - 6*x - 3. - Amiram Eldar, May 28 2026

Example

1.895643923738960001647011798432002122246...

Mathematica

FromContinuedFraction[{3/2, 2, {3/2, 2}}]
ContinuedFraction[%, 100]  (* [1, 1, 8, 1, 1, 2, ... *)
RealDigits[N[%%, 120]]
N[%%%, 40]
(* Alternative: *)
RealDigits[(3+Sqrt[21])/4, 10, 120][[1]] (* Harvey P. Dale, Dec 13 2019 *)

Prog

(PARI) (3+sqrt(21))/4 \\ Charles R Greathouse IV, May 18 2026

Crossrefs

Cf. A188635.

Sequence in context: A199460 A245772 A388591 * A198119 A395940 A088612

Adjacent sequences: A190286 A190287 A190288 * A190290 A190291 A190292

Keyword

nonn,cons,changed

Author

Clark Kimberling, May 07 2011

Status

approved