|
| |
|
|
A042103
|
|
Denominators of continued fraction convergents to sqrt(575).
|
|
2
|
|
|
|
1, 1, 47, 48, 2255, 2303, 108193, 110496, 5191009, 5301505, 249060239, 254361744, 11949700463, 12204062207, 573336561985, 585540624192, 27508205274817, 28093745899009, 1319820516629231, 1347914262528240, 63323876592928271, 64671790855456511
(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 = 46 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
|
|
|
|
FORMULA
|
G.f.: -(x^2-x-1) / (x^4-48*x^2+1). - Colin Barker, Dec 01 2013
The following remarks assume an offset of 1.
Let alpha = ( sqrt(46) + sqrt(50) )/2 and beta = ( sqrt(46) - sqrt(50) )/2 be the roots of the equation x^2 - sqrt(46)*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)} ( 46 + 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) = 46*a(2*n) + a(2*n - 1). (End)
|
|
|
MATHEMATICA
|
|
|
|
PROG
|
(Magma) I:=[1, 1, 47, 48]; [n le 4 select I[n] else 48*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jan 14 2014
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,frac,easy
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
