|
| |
|
|
A099373
|
|
Twice Chebyshev's polynomials of the first kind, T(n,x), evaluated at 83/2.
|
|
3
|
|
|
|
2, 83, 6887, 571538, 47430767, 3936182123, 326655685442, 27108485709563, 2249677658208287, 186696137145578258, 15493529705424787127, 1285776269413111753283, 106703936831582850735362
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,1
|
|
|
COMMENTS
|
Used in A099372.
The proper and improper nonnegative solutions of the Pell equation x(n)^2 - 85*y(n)^2 = +4 are x(n) = a(n) and y(n) = 9*A097839(n), n >= 0. - Wolfdieter Lang, Jul 01 2013
|
|
|
LINKS
|
Indranil Ghosh, Table of n, a(n) for n = 0..520
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (83,-1).
|
|
|
FORMULA
|
a(n) = 83*a(n-1) - a(n-2), n >= 1; a(-1) = 83, a(0) = 2.
a(n) = S(n, 83) - S(n-2, 83) = 2*T(n, 83/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 83)= A097839(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.
G.f.: (2-83*x)/(1-83*x+x^2).
a(n) = ap^n + am^n, with ap := (83+9*sqrt(85))/2 and am := (83-9*sqrt(85))/2.
|
|
|
EXAMPLE
|
Pell equation: n=0: 2^2 - 85*0^2 = +4 (improper), n=1: 83^2 - 85*(9*1)^2 = +4, n=2: 6887^2 - 85*(9*83)^2 = +4. - Wolfdieter Lang, Jul 01 2013
|
|
|
CROSSREFS
|
Sequence in context: A225807 A232770 A317724 * A169601 A205643 A215263
Adjacent sequences: A099370 A099371 A099372 * A099374 A099375 A099376
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
Wolfdieter Lang, Oct 18 2004
|
|
|
STATUS
|
approved
|
| |
|
|
