A319034 Decimal expansion of the height that minimizes the total surface area of the four triangular faces of a square pyramid of unit volume.

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

Offset

1,3

Comments

A square pyramid with a height of h and a base of size s X s has volume V = (1/3)*s^2*h, so a square pyramid of unit volume has s = sqrt(3/h), and the slant height of each of the four triangular faces is t = sqrt(h^2 + (s/2)^2) = sqrt(h^2 + 3/(4*h)), and the total area of the four faces is A = 4*(s*t/2) = sqrt(12*h^3 + 9)/h; this area is minimized at h = (3/2)^(1/3), where it reaches A = 3^(7/6)*2^(1/3).

If the total surface area of all five faces including the square base is to be minimized, then the resulting height is 6^(1/3) (cf. A005486). - Jon E. Schoenfield, Nov 11 2018

Links

Table of n, a(n) for n=1..100.

Eric Weisstein's World of Mathematics, Pyramid.

Formula

Equals (3/2)^(1/3) = (1/2)*A010584.

Equals A002581/A002580. - Michel Marcus, Oct 23 2018

Example

1.14471424255333186780804221193967700891590692078793...

Mathematica

RealDigits[Surd[3/2, 3], 10, 120][[1]] (* Amiram Eldar, Jun 21 2023 *)

Prog

(PARI) sqrtn(3/2, 3) \\ Michel Marcus, Oct 23 2018

Crossrefs

Cf. A002580, A002581, A005486, A010584.

Sequence in context: A198138 A300709 A240924 * A282468 A196463 A021695

Adjacent sequences: A319031 A319032 A319033 * A319035 A319036 A319037

Keyword

nonn,cons

Author

Jon E. Schoenfield, Oct 22 2018

Status

approved