A243469 Denominators of the rational convergents to the periodic continued fraction 1/(2 + 1/(7 + 1/(2 + 1/(7 + ...)))).
1, 2, 15, 32, 239, 510, 3809, 8128, 60705, 129538, 967471, 2064480, 15418831, 32902142, 245733825, 524369792, 3916322369, 8357014530, 62415424079, 133187862688, 994730462895, 2122648788478, 15853271982241, 33829192752960, 252657621252961, 539144435258882
Offset: 1
Links
- Peter Bala, Notes on 2-periodic continued fractions and Lehmer sequences
- Leonhard Euler, Introductio in analysin infinitorum, Vol.1, Chapter 18, section 378. French and German translations.
- Eric W. Weisstein, MathWorld: Lehmer Number
- Index to divisibility sequences
Programs
-
PARI
Vec(x*(1+2*x-x^2)/(1-16*x^2+x^4)+O(x^99)) \\ Charles R Greathouse IV, Nov 13 2015
Formula
Let alpha = ( sqrt(14) + sqrt(18) )/2 and beta = ( sqrt(14) - sqrt(18) )/2 be the roots of the equation x^2 - sqrt(14)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = 2*(alpha^n - beta^n)/(alpha^2 - beta^2) for n even.
a(2*n + 1) = Product_{k = 1..n} (14 + 4*cos^2(k*Pi/(2*n+1)));
a(2*n) = 2*Product_{k = 1..n-1} (14 + 4*cos^2(k*Pi/(2*n))).
Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 2, a(2*n) = 2*a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 7*a(2*n) + a(2*n - 1).
Fourth-order recurrence: a(n) = 16*a(n - 2) - a(n - 4) for n >= 5.
O.g.f.: x*(1 + 2*x - x^2)/(1 - 16*x^2 + x^4).
Comments