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

Author

Roman Witula, Oct 23 2012

Keywords

• nonn
• cons

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).

Examples

			0.8669386054934201...
		

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.

Crossrefs

Cf. A075778, A060006, A001608.

Programs

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 *)

PARI

asin(1/2/real(polroots(x^3+x^2-1)[1])^1.5) \\ Charles R Greathouse IV, Dec 11 2013

Formula

Equals arccos((1-A060006)/2)/2. - Gerry Martens, Apr 16 2024

Extensions

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