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A098247 First differences of Chebyshev polynomials S(n,227)=A098245(n) with Diophantine property. 5
1, 226, 51301, 11645101, 2643386626, 600037119001, 136205782626601, 30918112619119426, 7018275358757483101, 1593117588325329544501, 361630674274491049118626, 82088569942721142820383601 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

(15*b(n))^2 - 229*a(n)^2 = -4 with b(n)=A098246(n) give all positive solutions of this Pell equation.

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..423

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (227, -1).

Index entries for sequences related to Chebyshev polynomials.

FORMULA

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 Deléham, Nov 18 2008]

EXAMPLE

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), ...

CROSSREFS

Sequence in context: A251507 A251500 A050847 * A092994 A031513 A078765

Adjacent sequences:  A098244 A098245 A098246 * A098248 A098249 A098250

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Sep 10 2004

STATUS

approved

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Last modified April 26 03:45 EDT 2017. Contains 285426 sequences.