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Search: id:A098247
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| A098247 |
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First differences of Chebyshev polynomials S(n,227)=A098245(n) with Diophantine property. |
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5
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| 1, 226, 51301, 11645101, 2643386626, 600037119001, 136205782626601, 30918112619119426, 7018275358757483101, 1593117588325329544501, 361630674274491049118626, 82088569942721142820383601
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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(15*b(n))^2 - 229*a(n)^2 = -4 with b(n)=A098246(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)= S(n, 227) - S(n-1, 227) = T(2*n+1, sqrt(229)/2)/(sqrt(229)/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 second kind, A053120.
a(n)= ((-1)^n)*S(2*n, 15*I) with the imaginary unit I and the S(n, x)=U(n, x/2) Chebyshev polynomials.
G.f.: (1-x)/(1-227*x+x^2).
a(n)=227*a(n-1)-a(n-2), n>1 ; a(0)=1, a(1)=226 . [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 - 229*y^2 = -4 are
(15=15*1,1), (3420=15*228,226), (776325=15*51755,51301),
(176222355=15*11748157,11645101), ...
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CROSSREFS
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Sequence in context: A031708 A031603 A050847 this_sequence A092994 A031513 A078765
Adjacent sequences: A098244 A098245 A098246 this_sequence A098248 A098249 A098250
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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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