A086178 -id:A086178 - cp's OEIS frontend

A365823 Decimal expansion of 2*(2 + sqrt(2)).

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

Offset: 1

Wolfdieter Lang, Nov 13 2023

The greater one of the solutions to x^2 - 8 * x + 8 = 0. The other solution is A157259 - 3 = 1.17157... . - Michal Paulovic, Nov 14 2023

			6.8284271247461900976033774484193961571393437507538961...

Cf. A000129, A002193, A014176, A049310, A057084, A157259.
Cf. A156035, A163960.
Essentially the same as A157258, A090488, A086178 and A010466.

evalf(4+sqrt(8), 130);  # Alois P. Heinz, Nov 13 2023

First[RealDigits[2*(2 + Sqrt[2]), 10, 99]] (* Stefano Spezia, Nov 11 2023 *)

\\ Works in v2.13 and higher; n = 100 decimal places
my(n=100); digits(floor(10^n*(4+quadgen(32)))) \\ Michal Paulovic, Nov 14 2023

Equals 2*sqrt(2)*(1 + sqrt(2)) = 2*(2 + sqrt(2)). This is an integer in the quadratic number field Q(sqrt(2)).
Equals lim_{n->oo} A057084(n + 1)/A057084(n).
Equals continued fraction with periodic term [[6], [1, 4]]. - Peter Luschny, Nov 13 2023
Equals -3+A157258 = 1+A156035 = 2+A090488 = 3+A086178 = 4+A010466 = 6+A163960. - Alois P. Heinz, Nov 15 2023

A381756 Decimal expansion of the smallest angular distance between two vertices of the equilateral square antiprism measured along the circumscribing sphere.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Mar 06 2025

The equilateral square antiprism of side number n=4, lateral edge length a, and the two bases separated vertically by h has h = a*sqrt( 1-sec^2(Pi/(2n)) ) = a/2^(1/4). The 4 vertices of the top base have Cartesian coordinates (+-a/sqrt(2),0,h/2), (0,+-a/sqrt(2),h/2); the 4 vertices at the bottom base have (+-a/2,+-a/2,-h/2). The common distance of these 8 vertices from the origin is r = a*sqrt(8+2^(3/2))/4, the radius of the circumscribing sphere. The largest dot product between pairs of the 8 vertices is sqrt(2)*a^2/8 , which is equivalent to the smallest distance measured along the surface of the sphere of radius r. Dividing this dot product through r^2 gives 2^(3/2)/(8+2^(3/2)), the cosine of the angle between closest vertices. This here is the angle measured in radians.

			1.3065271617174372755341...

Eric Weisstein's World of Mathematics, Antiprism.

Cf. A086178.

Maple
```
evalf( arccos(1/(2^(3/2)+1)) ) ;
```

RealDigits[ArcCos[1/(2^(3/2)+1)],10,91][[1]] (* Stefano Spezia, Jul 29 2025 *)

Equals arccos(1/(2^(3/2)+1)) = arcsec(A086178).

cp's OEIS Frontend

A365823 Decimal expansion of 2*(2 + sqrt(2)).

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