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A097838 First differences of Chebyshev polynomials S(n,51)=A097836(n) with Diophantine property. 5
1, 50, 2549, 129949, 6624850, 337737401, 17217982601, 877779375250, 44749530155149, 2281348258537349, 116304011655249650, 5929223246159194801, 302274081542463685201, 15410048935419488750450, 785610221624851462587749 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

(7*b(n))^2 - 53*a(n)^2 = -4 with b(n)=A097837(n) give all positive solutions of this Pell equation.

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..584

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (51, -1).

Index entries for sequences related to Chebyshev polynomials.

FORMULA

a(n) = ((-1)^n)*S(2*n, 7*I) with the imaginary unit I and the S(n, x)=U(n, x/2) Chebyshev polynomials.

G.f.: (1-x)/(1-51*x+x^2).

a(n) = S(n, 51) - S(n-1, 51) = T(2*n+1, sqrt(53)/2)/(sqrt(53)/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) = 51*a(n-1)-a(n-2) ; a(0)=1, a(1)=50. [Philippe Deléham, Nov 18 2008]

EXAMPLE

All positive solutions of Pell equation x^2 - 53*y^2 = -4 are (7=7*1,1), (364=7*52,50), (18557=7*2651,2549), (946043=7*135149,129949), ...

CROSSREFS

Sequence in context: A223796 A165800 A042201 * A203842 A251058 A239653

Adjacent sequences:  A097835 A097836 A097837 * A097839 A097840 A097841

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Sep 10 2004

STATUS

approved

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Last modified August 23 04:06 EDT 2017. Contains 290958 sequences.