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A098244
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First differences of Chebyshev polynomials S(n,171)=A097844(n) with Diophantine property.
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4
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1, 170, 29069, 4970629, 849948490, 145336221161, 24851643870041, 4249485765555850, 726637214266180309, 124250714153751276989, 21246145483077202184810, 3632966626892047822325521
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| (13*b(n))^2 - 173*a(n)^2 = -4 with b(n)=A097845(n) give all positive solutions of this Pell equation.
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LINKS
| Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| a(n)= ((-1)^n)*S(2*n, 13*I) with the imaginary unit I and the S(n, x)=U(n, x/2) Chebyshev polynomials.
G.f.: (1-x)/(1-171*x+x^2).
a(n)= S(n, 171) - S(n-1, 171) = T(2*n+1, sqrt(173)/2)/(sqrt(173)/2), with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x)= 0 = U(-1, x) and T(n, x) Chebyshev's polynomials of the first kind, A053120.
a(n)=171*a(n-1)-a(n-2), n>1 ; a(0)=1, a(1)=170 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 18 2008]
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EXAMPLE
| All positive solutions of Pell equation x^2 - 173*y^2 = -4 are
(13=13*1,1), (2236=13*172,170), (382343=13*29411,29069),
(65378417=13*5029109,4970629), ...
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CROSSREFS
| Sequence in context: A187520 A178499 A133328 * A114048 A187704 A185239
Adjacent sequences: A098241 A098242 A098243 * A098245 A098246 A098247
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Sep 10 2004
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