A218197 Decimal expansion of the Perrin argument a (see below).

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

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.

Links

Table of n, a(n) for n=0..119.

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

Cf. A075778, A060006, A001608.

Sequence in context: A154499 A329080 A092750 * A181048 A199873 A300503

Adjacent sequences: A218194 A218195 A218196 * A218198 A218199 A218200

Keyword

nonn,cons,changed

Author

Roman Witula, Oct 23 2012

Extensions

a(119) corrected by Sean A. Irvine, Apr 16 2024

Status

approved