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%I #38 Feb 25 2023 03:07:46
%S 1,65,4289,283009,18674305,1232221121,81307919681,5365090477825,
%T 354014663616769,23359602708228929,1541379764079492545,
%U 101707704826538279041,6711167138787446924161,442835323455144958715585,29220420180900779828304449,1928104896615996323709378049
%N Chebyshev sequence with Diophantine property.
%C Bisection (even part) of A041025.
%C (4*A078989(n))^2 - 17*a(n)^2 = -1 (Pell -1 equation, see A077232-3).
%C Starting with a(1), hypotenuses of primitive Pythagorean triples in A195619 and A195620. - _Clark Kimberling_, Sep 22 2011
%H Colin Barker, <a href="/A078988/b078988.txt">Table of n, a(n) for n = 0..549</a>
%H A. J. C. Cunningham, <a href="https://archive.org/details/binomialfactoris01cunn/page/n46/mode/1up">Binomial Factorisations</a>, Vols. 1-9, Hodgson, London, 1923-1929. See Vol. 1, page xxxv.
%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 (66,-1).
%F G.f.: (1-x)/(1-66*x+x^2).
%F a(n) = T(2*n+1, sqrt(17))/sqrt(17) = ((-1)^n)*S(2*n, 8*i) = S(n, 66) - S(n-1, 66) with i^2=-1 and T(n, x), resp. S(n, x), Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310.
%F a(n) = A041025(2*n).
%F a(n) = 66*a(n-1) - a(n-2) for n>1 ; a(0)=1, a(1)=65. - _Philippe Deléham_, Nov 18 2008
%e (x,y) = (4,1), (268,65), (17684,4289), ... give the positive integer solutions to x^2 - 17*y^2 =-1.
%t CoefficientList[Series[(1-x)/(1-66x+x^2), {x,0,20}], x] (* _Michael De Vlieger_, Apr 15 2019 *)
%t LinearRecurrence[{66,-1}, {1,65}, 21] (* _G. C. Greubel_, Aug 01 2019 *)
%o (PARI) Vec((1-x)/(1-66*x+x^2) + O(x^20)) \\ _Colin Barker_, Jun 15 2015
%o (Magma) I:=[1, 65]; [n le 2 select I[n] else 66*Self(n-1) - Self(n-2): n in [1..20]]; // _G. C. Greubel_, Aug 01 2019
%o (Sage) ((1-x)/(1-66*x+x^2)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 01 2019
%o (GAP) a:=[1,65];; for n in [3..20] do a[n]:=66*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Aug 01 2019
%Y Row 66 of array A094954.
%Y Cf. A097316 for S(n, 66).
%Y Row 4 of array A188647.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Jan 10 2003
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