OFFSET
1,2
COMMENTS
The sequence of convergents to the simple periodic continued fraction 1/(2 + 1/(7 + 1/(2 + 1/(7 + ...)))) begins [0/1, 1/2, 7/15, 15/32, 112/239, 239/510, ...]. Euler considers these convergents, in section 378 of the first volume of his textbook Introductio in Analysin Infinitorum, as a way of finding rational approximations to sqrt(7). The present sequence is the sequence of denominators of the convergents. It is a strong divisibility sequence, that is gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. The sequence is closely related to A041111, the Lehmer numbers U_n(sqrt(R),Q)) with parameters R = 14 and Q = -1.
See A243470 for the sequence of numerators to the convergents.
LINKS
L. Euler, Introductio in analysin infinitorum, Vol.1, Chapter 18, section 378. French and German translations.
Eric W. Weisstein, MathWorld: Lehmer Number
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).
PROG
(PARI) Vec(x*(1+2*x-x^2)/(1-16*x^2+x^4)+O(x^99)) \\ Charles R Greathouse IV, Nov 13 2015
CROSSREFS
KEYWORD
nonn,easy,frac,changed
AUTHOR
Peter Bala, Jun 06 2014
STATUS
approved