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Search: id:A098253
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| A098253 |
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First differences of Chebyshev polynomials S(n,363)=A098251(n) with Diophantine property. |
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4
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| 1, 362, 131405, 47699653, 17314842634, 6285240176489, 2281524869222873, 828187242287726410, 300629687425575463957, 109127748348241605689981, 39613072020724277289999146, 14379436015774564414664000017
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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(19*b(n))^2 - 365*a(n)^2 = -4 with b(n)=A098252(n) give all positive solutions of this Pell equation.
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LINKS
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Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
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a(n)= ((-1)^n)*S(2*n, 19*I) with the imaginary unit I and the S(n, x)=U(n, x/2) Chebyshev polynomials.
G.f.: (1-x)/(1-363*x+x^2).
a(n)= S(n, 363) - S(n-1, 363) = T(2*n+1, sqrt(365)/2)/(sqrt(365)/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)=363*a(n-1)-a(n-2), n>1 ; a(0)=1, a(1)=362 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 18 2008]
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EXAMPLE
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All positive solutions of Pell equation x^2 - 365*y^2 = -4 are
(19=19*1,1), (6916=19*364,362), (2510489=19*132131,131405),
(911300591=19*47963189,47699653), ...
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CROSSREFS
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Sequence in context: A020539 A157442 A007565 this_sequence A031517 A116285 A031697
Adjacent sequences: A098250 A098251 A098252 this_sequence A098254 A098255 A098256
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Sep 10 2004
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