A193537 Decimal expansion of cos(Pi/(1+phi)), where phi is the golden ratio.

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

Offset

0,1

Comments

cos(Pi/(1+phi)) is the first term in the identity:

cos(Pi/(1+phi))+cos(Pi/phi)=0 which when converted to the exponential form gives: e^(i*Pi/(1+phi))+e^(-i*Pi/(1+phi))+e^(i*Pi/phi)+e^(-i*Pi/phi)=0. In this form it is known as the phi identity because it combines the golden ratio phi with the five fundamental mathematical constants Pi, e, i, 1, 0 that are found in Euler's identity e^(i*Pi) + 1 = 0.

Links

Table of n, a(n) for n=0..98.

Frank M. Jackson, Five a day (Letter to Editor), Mathematics Today 50-6 (2014) 321.

Formula

c = cos(Pi/(1+phi)) = cos(2*A180014).

Example

0.3623748900804801199586466374749868993608655440055985464501567887401235062...

Mathematica

N[Cos[Pi/(1+GoldenRatio)], 100]

Prog

(PARI) cos((3-sqrt(5))*Pi/2) \\ Charles R Greathouse IV, Jul 29 2011

Crossrefs

Sequence in context: A132120 A021280 A282728 * A319674 A021738 A016614

Adjacent sequences: A193534 A193535 A193536 * A193538 A193539 A193540

Keyword

easy,nonn,cons

Author

Frank M Jackson, Jul 29 2011

Status

approved