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

%I A218197 #45 Apr 16 2024 10:26:26
%S A218197 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,
%T A218197 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,
%U A218197 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
%N A218197 Decimal expansion of the Perrin argument a (see below).
%C A218197 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).
%C A218197 The Perrin polynomial is the characteristic polynomial of the Perrin recurrence sequence (see A001608):
%C A218197 A(n) = A(n-2) + A(n-3), with A(0)=3, A(1)=0, and A(2)=2.
%C A218197 The Binet formula of this sequence has the form
%C A218197 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),
%C A218197 which implies the relations
%C A218197 A(2*n) = t^(-2*n) + 2 * (-1)^n * cos(2*a*n) * t^n, and
%C A218197 A(2*n-1) = t^(-2*n+1) + 2 * (-1)^(n-1) * sin((2*n-1)*a) * t^(n - 1/2).
%C A218197 It is proved in the paper of Witula et al. that we have
%C A218197 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).
%D A218197 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.
%F A218197 Equals arccos((1-A060006)/2)/2. - _Gerry Martens_, Apr 16 2024
%e A218197 0.8669386054934201...
%t A218197 ArcSin[1/(2*Root[Function[x, x^3+x^2-1], 1]^(3/2))] // RealDigits[#, 10, 120]& // First (* _Jean-François Alcover_, Feb 20 2014 *)
%o A218197 (PARI) asin(1/2/real(polroots(x^3+x^2-1)[1])^1.5) \\ _Charles R Greathouse IV_, Dec 11 2013
%Y A218197 Cf. A075778, A060006, A001608.
%K A218197 nonn,cons
%O A218197 0,1
%A A218197 _Roman Witula_, Oct 23 2012
%E A218197 a(119) corrected by _Sean A. Irvine_, Apr 16 2024