A243967 Decimal expansion of 'xi', a constant related to the second order quadratic recurrence q(0)=q(1)=1, q(n)=q(n-2)*(q(n-1)+1).

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

Offset

1,2

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.10 Quadratic recurrence constants, pp. 445-446.

Links

Table of n, a(n) for n=1..99.

Formula

q(n) = floor(xi^(phi^n)*eta^((1-phi)^n)) where phi is the golden ratio (1+sqrt(5))/2 and eta is A243968.

Example

1.3505061251311715303318376772262415972523...

Mathematica

digits = 99; n0 = 5; dn = 5; Clear[q]; q[0] = q[1] = 1; q[n_] := q[n] = q[n - 2] (q[n - 1] + 1); xi[n_] := xi[n] = ((q[n] - 1)^(1/2*( Sqrt[5] - 1))*(q[n + 1] - 1))^((1/2*( Sqrt[5] - 1))^n/Sqrt[5]); xi[n0]; xi[n = n0 + dn]; While[RealDigits[xi[n], 10, digits + 10] != RealDigits[xi[n - 5], 10, digits + 10], Print["n = ", n];  n = n + dn]; RealDigits[xi[n], 10, digits] // First

Crossrefs

Cf. A006277, A243968 (eta).

Sequence in context: A200520 A224933 A307209 * A144541 A100609 A104866

Adjacent sequences: A243964 A243965 A243966 * A243968 A243969 A243970

Keyword

nonn,cons

Author

Jean-François Alcover, Jun 16 2014

Status

approved