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A042623 Denominators of continued fraction convergents to sqrt(840). 2
1, 1, 57, 58, 3305, 3363, 191633, 194996, 11111409, 11306405, 644270089, 655576494, 37356553753, 38012130247, 2166035847585, 2204047977832, 125592722606177, 127796770584009, 7282211875310681, 7410008645894690, 422242696045413321 (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 = 56 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 27 2014

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Eric W. Weisstein, MathWorld: Lehmer Number

Index entries for linear recurrences with constant coefficients, signature (0, 58, 0, -1).

FORMULA

G.f.: -(x^2-x-1) / (x^4-58*x^2+1). - Colin Barker, Dec 20 2013

From Peter Bala, May 27 2014: (Start)

The following remarks assume an offset of 1.

Let alpha = sqrt(14) + sqrt(15) and beta = sqrt(14) - sqrt(15) be the roots of the equation x^2 - sqrt(56)*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)} ( 56 + 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) = 56*a(2*n) + a(2*n - 1). (End)

MATHEMATICA

Denominator[Convergents[Sqrt[840], 30]] (* Vincenzo Librandi, Jan 26 2014 *)

LinearRecurrence[{0, 58, 0, -1}, {1, 1, 57, 58}, 30] (* Harvey P. Dale, Oct 31 2016 *)

CROSSREFS

Cf. A042622, A040811. A002530.

Sequence in context: A036184 A217923 A136542 * A072466 A216183 A334238

Adjacent sequences:  A042620 A042621 A042622 * A042624 A042625 A042626

KEYWORD

nonn,frac,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Colin Barker, Dec 20 2013

STATUS

approved

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Last modified August 15 12:56 EDT 2020. Contains 336502 sequences. (Running on oeis4.)