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A098292
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First differences of Chebyshev polynomials S(n,731)=A098263(n) with Diophantine property.
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3
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1, 730, 533629, 390082069, 285149458810, 208443864308041, 152372179659719161, 111383854887390398650, 81421445550502721693989, 59518965313562602167907309, 43508282222768711682018548890
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| (27*b(n))^2 - 733*a(n)^2 = -4 with b(n)=A098291(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, 27*I) with the imaginary unit I and the S(n, x)=U(n, x/2) Chebyshev polynomials.
G.f.: (1-x)/(1-731*x+x^2).
a(n)= S(n, 731) - S(n-1, 731) = T(2*n+1, sqrt(733)/2)/(sqrt(733)/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)=731*a(n-1)-a(n-2), n>1 ; a(0)=1, a(1)=730 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 18 2008]
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EXAMPLE
| All positive solutions of Pell equation x^2 - 733*y^2 = -4 are
(27=27*1,1), (19764=27*732,730), (14447457=27*535091,533629),
(10561071303=27*391150789,390082069), ...
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CROSSREFS
| Sequence in context: A031732 A031615 A085441 * A031525 A031705 A158396
Adjacent sequences: A098289 A098290 A098291 * A098293 A098294 A098295
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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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