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A098246 Chebyshev polynomials S(n,227) + S(n-1,227) with Diophantine property. 4
1, 228, 51755, 11748157, 2666779884, 605347285511, 137411167031113, 31191729568777140, 7080385200945379667, 1607216248885032407269, 364831008111701411070396, 82815031625107335280572623 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

(15*a(n))^2 - 229*b(n)^2 = -4 with b(n)=A098247(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 sequences related to Chebyshev polynomials.

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

FORMULA

a(n)= S(n, 227) + S(n-1, 227) = S(2*n, sqrt(229)), with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x)= 0 = U(-1, x). S(n, 227)=A098245(n).

a(n)= (-2/15)*I*((-1)^n)*T(2*n+1, 15*I/2) with the imaginary unit I and Chebyshev's polynomials of the first kind. See the T-triangle A053120.

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)=228 . [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), ...

MATHEMATICA

LinearRecurrence[{227, -1}, {1, 228}, 20] (* Harvey P. Dale, May 29 2014 *)

CROSSREFS

Sequence in context: A064245 A201238 A220624 * A091551 A033528 A086002

Adjacent sequences:  A098243 A098244 A098245 * A098247 A098248 A098249

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Sep 10 2004

STATUS

approved

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Last modified June 28 05:58 EDT 2017. Contains 288813 sequences.