A380003 - OEIS

%I #9 Jan 13 2025 04:13:42

%S 1,1,7,7,2,8,5,8,2,3,4,7,1,7,5,0,2,9,1,9,2,3,5,3,7,4,4,5,4,8,1,2,4,4,

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

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

%N Decimal expansion of acute vertex angle, in radians, in a pentagonal hexecontahedron face.

%C A pentagonal hexecontahedron face is an irregular pentagon with one acute angle (this constant) and four (equal) obtuse angles (A380004).

%H Paolo Xausa, <a href="/A380003/b380003.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalHexecontahedron.html">Pentagonal Hexecontahedron</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Pentagonal_hexecontahedron">Pentagonal hexecontahedron</a>.

%F Equals arccos(c), where c is the largest real root of 64*x^6 - 384*x^5 + 384*x^4 + 888*x^3 + 168*x^2 - 128*x - 31.

%F Equals 3*Pi - 4*A380004.

%e 1.1772858234717502919235374454812446809073054345981...

%t First[RealDigits[ArcCos[Root[64*#^6 - 384*#^5 + 384*#^4 + 888*#^3 + 168*#^2 - 128*# - 31 &, 4]], 10, 100]]

%Y Cf. A379888 (surface area), A379889 (volume), A379890 (inradius), A379890 (midradius), A379892 (dihedral angle), A380002 (long/short edge length ratio), A380004 (face obtuse angles).

%K nonn,cons,easy

%O 1,3

%A _Paolo Xausa_, Jan 12 2025