|
Search: id:A097782
|
|
|
| A097782 |
|
Chebyshev polynomials S(n,29) with diophantine property. |
|
+0
2
|
|
| 1, 29, 840, 24331, 704759, 20413680, 591291961, 17127053189, 496093250520, 14369577211891, 416221645894319, 12056058153723360, 349209464812083121, 10115018421396687149, 292986324755691844200, 8486488399493666794651
(list; graph; listen)
|
|
|
OFFSET
|
0,2
|
|
|
COMMENT
|
All positive integer solutions of Pell equation b(n)^2 - 837*a(n)^2 = +4 together with b(n)=A090251(n+1), n>=0. Note that D=837=93*3^2 is not square-free.
|
|
LINKS
|
Tanya Khovanova, Recursive Sequences
Index entries for sequences relate d to Chebyshev polynomials.
|
|
FORMULA
|
a(n)= S(n, 29)=U(n, 29/2)= S(2*n+1, sqrt(31))/sqrt(31) with S(n, x)=U(n, x/2) Chebyshev's polynomials of the 2nd kind, A049310. S(-1, x)= 0 = U(-1, x).
a(n)=29*a(n-1)-a(n-2), n >= 1; a(0)=1, a(1)=29; a(-1)=0.
a(n)=(ap^(n+1) - am^(n+1))/(ap-am) with ap := (29+3*sqrt(93))/2 and am := (29-3*sqrt(93))/2.
G.f.: 1/(1-29*x+x^2).
|
|
EXAMPLE
|
(x,y) = (29;1), (839;29), (24302,840), ..., give the positive integer solutions to x^2 - 93*(3*y)^2 =+4.
|
|
CROSSREFS
|
Adjacent sequences: A097779 A097780 A097781 this_sequence A097783 A097784 A097785
Sequence in context: A062560 A135995 A046850 this_sequence A009973 A057687 A049667
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 31 2004
|
|
|
Search completed in 0.002 seconds
|
