A189964 Decimal expansion of (3+x+sqrt(38+6*x))/4, where x = sqrt(13).

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

Offset

1,1

Comments

This constant is the shape of a rectangle whose continued fraction partition matches [r,r,r,...], where r = (3+sqrt(13))/2. For a general discussion, see A188635. The ordinary continued fraction of r is [3,3,3,3,3,3,3,3,3,3,...]. A rectangle of shape r (that is, (length/width)=r) may be compared with the golden rectangle, with shape [1,1,1,1,1,1,...], and the silver rectangle, with shape [2,2,2,2,2,2,...].

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 - 3*x^3 - 3*x^2 + 3*x + 1. - Amiram Eldar, Jun 01 2026

Example

3.5819529507118503770725396959210446869118915483494611612922...

Mathematica

r = (3 +13^(1/2))/2;
FromContinuedFraction[{r, {r}}]
FullSimplify[%]
N[%, 150]
RealDigits[%]  (*A189964*)
ContinuedFraction[%%, 120] (*A189965*)

Prog

(PARI) (3+sqrt(13)+sqrt(38+6*sqrt(13)))/4 \\ G. C. Greubel, Jan 12 2018
(Magma) (3+Sqrt(13)+Sqrt(38+6*Sqrt(13)))/4; // G. C. Greubel, Jan 12 2018

Crossrefs

Cf. A188635, A188636, A189965 (continued fraction).

Sequence in context: A198838 A374774 A089103 * A181910 A394002 A367453

Adjacent sequences: A189961 A189962 A189963 * A189965 A189966 A189967

Keyword

nonn,cons,easy,changed

Author

Clark Kimberling, May 04 2011

Status

approved