A188656 Decimal expansion of (1+sqrt(65))/8.

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

Offset

1,3

Comments

Apart from the second digit the same as A177707.

Decimal expansion of the shape of a (1/4)-extension rectangle.

See A188640 for definitions of shape and r-extension rectangle.

A (1/4)-extension rectangle matches the continued fraction [1,7,1,1,7,1,1,7,1,1,7,1,1,7,...] for the shape L/W = (1+sqrt(65))/8. This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,...]. Specifically, for the (4/3)-extension rectangle, 1 square is removed first, then 7 squares, then 1 square, then 1 square, then 7 squares,..., so that the original rectangle is partitioned into an infinite collection of squares.

Links

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Clark Kimberling, A Visual Euclidean Algorithm, The Mathematics Teacher 76 (1983), 108-109.

Index entries for algebraic numbers, degree 2.

Formula

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

Example

1.13278221853731870654582665....

Mathematica

RealDigits[(1 + Sqrt[65])/8, 10, 111][[1]] (* Robert G. Wilson v, Aug 19 2011 *)

Prog

(PARI) (1+sqrt(65))/8 \\ Charles R Greathouse IV, May 18 2026

Crossrefs

Cf. A188640, A105395, A177707.

Sequence in context: A226370 A054183 A357939 * A210204 A208657 A329940

Adjacent sequences: A188653 A188654 A188655 * A188657 A188658 A188659

Keyword

nonn,cons,changed

Author

Clark Kimberling, Apr 09 2011

Status

approved