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A098292 First differences of Chebyshev polynomials S(n,731)=A098263(n) with Diophantine property. 4
1, 730, 533629, 390082069, 285149458810, 208443864308041, 152372179659719161, 111383854887390398650, 81421445550502721693989, 59518965313562602167907309, 43508282222768711682018548890 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
(27*b(n))^2 - 733*a(n)^2 = -4 with b(n)=A098291(n) give all positive solutions of this Pell equation.
LINKS
Tanya Khovanova, Recursive Sequences
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
FORMULA
a(n) = ((-1)^n)*S(2*n, 27*i) with the imaginary unit i and the S(n, x) = U(n, x/2) Chebyshev polynomials.
G.f.: (1-x)/(1-731*x+x^2).
a(n) = S(n, 731) - S(n-1, 731) = T(2*n+1, sqrt(733)/2)/(sqrt(733)/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) = 731*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=730. - Philippe Deléham, Nov 18 2008
EXAMPLE
All positive solutions of Pell equation x^2 - 733*y^2 = -4 are (27=27*1,1), (19764=27*732,730), (14447457=27*535091,533629), (10561071303=27*391150789,390082069), ...
MATHEMATICA
LinearRecurrence[{731, -1}, {1, 730}, 20] (* Harvey P. Dale, Nov 15 2013 *)
PROG
(PARI) my(x='x+O('x^20)); Vec((1-x)/(1-731*x+x^2)) \\ G. C. Greubel, Aug 01 2019
(Magma) I:=[1, 730]; [n le 2 select I[n] else 731*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
(Sage) ((1-x)/(1-731*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
(GAP) a:=[1, 730];; for n in [3..20] do a[n]:=731*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019
CROSSREFS
Cf. A098291.
Sequence in context: A224437 A259322 A085441 * A031525 A031705 A158396
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Sep 10 2004
STATUS
approved

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Last modified May 29 11:24 EDT 2024. Contains 372940 sequences. (Running on oeis4.)