OFFSET
0,1
COMMENTS
The Perrin argument a is defined by the decomposition of the known Perrin polynomial: X^3 - X - 1 = (X - t^(-1))*(X - i*sqrt(t)*e^(i*a))*(X + i*sqrt(t)*e^(-i*a)), where t = 0.754877666... (see A075778 and A060006 for the decimal expansions of t and t^(-1) respectively) is the only positive root of the polynomial x^3 + x^2 - 1 and a := arcsin(1/(2*sqrt(t^3))) (the principal value of arc sine is considered here).
The Perrin polynomial is the characteristic polynomial of the Perrin recurrence sequence (see A001608):
A(n) = A(n-2) + A(n-3), with A(0)=3, A(1)=0, and A(2)=2.
The Binet formula of this sequence has the form
A(n) = t^(-n) + i^n * t^(n/2) * (e^(i*(a + Pi)*n) + e^(-i*a*n)) = t^(-n) + 2*(-1)^n*t^(n/2)*cos((a + Pi/2)*n),
which implies the relations
A(2*n) = t^(-2*n) + 2 * (-1)^n * cos(2*a*n) * t^n, and
A(2*n-1) = t^(-2*n+1) + 2 * (-1)^(n-1) * sin((2*n-1)*a) * t^(n - 1/2).
It is proved in the paper of Witula et al. that we have
u + v + w = 0 for the respective complex values of the roots: u in (1 + t^(-1))^(1/3), v in (1 + i*sqrt(t)*e^(i*a))^(1/3) and w in (1 - i*sqrt(t)*e^(-i*a))^(1/3).
REFERENCES
R. Witula, E. Hetmaniok, and D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, submitted to Proceedings of the 15th International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.
FORMULA
Equals arccos((1-A060006)/2)/2. - Gerry Martens, Apr 16 2024
EXAMPLE
0.8669386054934201...
MATHEMATICA
ArcSin[1/(2*Root[Function[x, x^3+x^2-1], 1]^(3/2))] // RealDigits[#, 10, 120]& // First (* Jean-François Alcover, Feb 20 2014 *)
PROG
(PARI) asin(1/2/real(polroots(x^3+x^2-1)[1])^1.5) \\ Charles R Greathouse IV, Dec 11 2013
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Roman Witula, Oct 23 2012
EXTENSIONS
a(119) corrected by Sean A. Irvine, Apr 16 2024
STATUS
approved