|
%I #31 Sep 08 2022 08:45:15
%S 1,730,533629,390082069,285149458810,208443864308041,
%T 152372179659719161,111383854887390398650,81421445550502721693989,
%U 59518965313562602167907309,43508282222768711682018548890
%N First differences of Chebyshev polynomials S(n,731)=A098263(n) with Diophantine property.
%C (27*b(n))^2 - 733*a(n)^2 = -4 with b(n)=A098291(n) give all positive solutions of this Pell equation.
%H G. C. Greubel, <a href="/A098292/b098292.txt">Table of n, a(n) for n = 0..340</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Giovanni Lucca, <a href="http://forumgeom.fau.edu/FG2019volume19/FG201902index.html">Integer Sequences and Circle Chains Inside a Hyperbola</a>, Forum Geometricorum (2019) Vol. 19, 11-16.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (731,-1).
%F 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.
%F G.f.: (1-x)/(1-731*x+x^2).
%F 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.
%F a(n) = 731*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=730. - _Philippe Deléham_, Nov 18 2008
%e 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), ...
%t LinearRecurrence[{731,-1},{1,730},20] (* _Harvey P. Dale_, Nov 15 2013 *)
%o (PARI) my(x='x+O('x^20)); Vec((1-x)/(1-731*x+x^2)) \\ _G. C. Greubel_, Aug 01 2019
%o (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
%o (Sage) ((1-x)/(1-731*x+x^2)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 01 2019
%o (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
%Y Cf. A098291.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Sep 10 2004
|
