|
|
A243470
|
|
Numerators of the rational convergents to the periodic continued fraction 1/(2 + 1/(7 + 1/(2 + 1/(7 + ...)))).
|
|
2
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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(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
|
(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))
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy,frac
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|