A243470 Numerators of the rational convergents to the periodic continued fraction 1/(2 + 1/(7 + 1/(2 + 1/(7 + ...)))).

1, 7, 15, 112, 239, 1785, 3809, 28448, 60705, 453383, 967471, 7225680, 15418831, 115157497, 245733825, 1835294272, 3916322369, 29249550855, 62415424079, 466157519408, 994730462895, 7429270759673, 15853271982241, 118402174635360, 252657621252961, 1887005523406087

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, ...]. The present sequence is the sequence of numerators 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 A243469 for the sequence of denominators to the convergents.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

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 Weisstein's World of Mathematics, Lehmer Number

Index to divisibility sequences

Index entries for linear recurrences with constant coefficients, signature (0,16,0,-1).

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) = 7*(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) = 7*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) = 7*a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 2*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 + 7*x - x^2)/(1 - 16*x^2 + x^4).

a(2n-1) = A157456(n), a(2n) = 7*A077412(n-1). - Ralf Stephan, Jun 13 2014

a(n) = (1/2)*( 7*(1+(-1)^n)*ChebyshevU((n-2)/2, 8) + (1-(-1)^n)*(ChebyshevU((n- 1)/2, 8) - ChebyshevU((n-3)/2, 8)) ). - G. C. Greubel, May 21 2022

Mathematica

LinearRecurrence[{0, 16, 0, -1}, {1, 7, 15, 112}, 30] (* Harvey P. Dale, Nov 06 2017 *)

Prog

(PARI) Vec(x*(1+7*x-x^2)/(1-16*x^2+x^4)+O(x^99)) \\ Charles R Greathouse IV, Nov 13 2015
(Magma) I:=[1, 7, 15, 112]; [n le 4 select I[n] else 16*Self(n-2) -Self(n-4): n in [1..31]]; // G. C. Greubel, May 21 2022
(SageMath)
def b(n): return chebyshev_U(n, 8) # b=A077412
def A243470(n): return 7*((n-1)%2)*b(n//2 -1) +(n%2)*(b((n-1)//2) -b((n-1)//2 -1))
[A243470(n) for n in (1..30)] # G. C. Greubel, May 21 2022

Crossrefs

Cf. A041111, A077412, A243469.

Sequence in context: A231466 A032004 A032018 * A068366 A156499 A137881

Adjacent sequences: A243467 A243468 A243469 * A243471 A243472 A243473

Keyword

nonn,easy,frac,changed

Author

Peter Bala, Jun 06 2014

Status

approved