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URL: https://oeis.org/A098253

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A098253
First differences of Chebyshev polynomials S(n,363) = A098251(n) with Diophantine property.
4
1, 362, 131405, 47699653, 17314842634, 6285240176489, 2281524869222873, 828187242287726410, 300629687425575463957, 109127748348241605689981, 39613072020724277289999146, 14379436015774564414664000017, 5219695660654146158245742007025, 1894735145381439280878789684550058
OFFSET
0,2
COMMENTS
(19*b(n))^2 - 365*a(n)^2 = -4 with b(n)=A098252(n) give all positive solutions of this Pell equation.
FORMULA
G.f.: (1 - x)/(1 - 363*x + x^2).
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.
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) for n>1, a(0)=1, a(1)=362. - Philippe Deléham, Nov 18 2008
EXAMPLE
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), ...
MATHEMATICA
LinearRecurrence[{363, -1}, {1, 362}, 20] (* G. C. Greubel, Aug 01 2019 *)
PROG
(PARI) my(x='x+O('x^20)); Vec((1-x)/(1-363*x+x^2)) \\ G. C. Greubel, Aug 01 2019
(Magma) I:=[1, 362]; [n le 2 select I[n] else 363*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
(SageMath) ((1-x)/(1-363*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
(GAP) a:=[1, 362];; for n in [3..20] do a[n]:=363*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019
CROSSREFS
Sequence in context: A200558 A212324 A007565 * A031517 A192449 A116285
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Sep 10 2004
STATUS
approved