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A041059 Denominators of continued fraction convergents to sqrt(35). 7
1, 1, 11, 12, 131, 143, 1561, 1704, 18601, 20305, 221651, 241956, 2641211, 2883167, 31472881, 34356048, 375033361, 409389409, 4468927451, 4878316860, 53252096051, 58130412911, 634556225161, 692686638072, 7561422605881, 8254109243953, 90102515045411 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 10 and Q = -1; it is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 28 2014
LINKS
Eric Weisstein's World of Mathematics, Lehmer Number
FORMULA
G.f.: (1+x-x^2)/(1-12*x^2+x^4). - Colin Barker, Jan 01 2012
From Peter Bala, May 28 2014: (Start)
The following remarks assume an offset of 1.
Let alpha = ( sqrt(10) + sqrt(14) )/2 and beta = ( sqrt(10) - sqrt(14) )/2 be the roots of the equation x^2 - sqrt(10)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even.
a(n) = Product_{k = 1..floor((n-1)/2)} ( 10 + 4*cos^2(k*Pi/n) ).
Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 1, a(2*n) = a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 10*a(2*n) + a(2*n - 1). (End)
MATHEMATICA
Denominator[Convergents[Sqrt[35], 30]] (* Vincenzo Librandi, Oct 23 2013 *)
CROSSREFS
Sequence in context: A038326 A249312 A357273 * A041260 A109665 A041261
KEYWORD
nonn,frac,easy
AUTHOR
STATUS
approved

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Last modified July 17 09:00 EDT 2024. Contains 374363 sequences. (Running on oeis4.)