

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


EXAMPLE

1.14471424255333186780804221193967700891590692078793...


PROG

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


CROSSREFS

Cf. A002580, A002581, A005486.
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



