A243968 Decimal expansion of 'eta', a constant related to the second order quadratic recurrence q(0)=q(1)=1, q(n)=q(n-2)*(q(n-1)+1).

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

Offset: 1

Author

Jean-François Alcover, Jun 16 2014

Keywords

• nonn
• cons

Examples

			1.42981549990099451970390644376276...
		

References

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

Crossrefs

Cf. A006277, A243967 (xi).

Programs

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

Formula

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