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A097841
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First differences of Chebyshev polynomials S(n,83)=A097839(n) with Diophantine property.
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4
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1, 82, 6805, 564733, 46866034, 3889316089, 322766369353, 26785719340210, 2222891938868077, 184473245206710181, 15309056460218076946, 1270467212952893676337, 105433469618629957059025
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| (9*b(n))^2 - 85*a(n)^2 = -4 with b(n)=A097840(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, 9*I) with the imaginary unit I and the S(n, x)=U(n, x/2) Chebyshev polynomials.
G.f.: (1-x)/(1-83*x+x^2).
a(n)= S(n, 83) - S(n-1, 83) = T(2*n+1, sqrt(85)/2)/(sqrt(85)/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)=83*a(n-1)-a(n-2) for n>1 ; a(0)=1, a(1)=82. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 18 2008]
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EXAMPLE
| All positive solutions of Pell equation x^2 - 85*y^2 = -4 are
(9=9*1,1), (756=9*84,82), (62739=9*6971,6805), (5206581=9*578509,564733), ...
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CROSSREFS
| Sequence in context: A035736 A017745 A203169 * A116123 A116142 A054214
Adjacent sequences: A097838 A097839 A097840 * A097842 A097843 A097844
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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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