|
%I #47 Sep 08 2022 08:44:47
%S 1,6,71,846,10081,120126,1431431,17057046,203253121,2421980406,
%T 28860511751,343904160606,4097989415521,48831968825646,
%U 581885636492231,6933795669081126,82623662392481281,984550153040694246,11731978174095849671,139799187936109501806
%N a(n) = 12*a(n-1) - a(n-2).
%C From _Wolfdieter Lang_, Nov 08 2002: (Start)
%C Chebyshev's polynomials T(n,x) evaluated at x=6.
%C a(n+1) give all (nontrivial, integer) solutions of Pell equation a(n+1)^2 - 35*b(n)^2 = +1 with b(n)=A004191(n), n>=0. (End)
%C a(35+70k)-1 and a(35+70k)+1 are consecutive odd powerful numbers. The first pair is 23101441813552306872262673994181386126 +- 1. See A076445. - _T. D. Noe_, May 04 2006
%C Numbers n such that 35*(n^2-1) is a square. - _Vincenzo Librandi_, Nov 19 2010
%C Except for the first term, positive values of x (or y) satisfying x^2 - 12xy + y^2 + 35 = 0. - _Colin Barker_, Feb 09 2014
%H T. D. Noe, <a href="/A023038/b023038.txt">Table of n, a(n) for n = 0..200</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%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 (12,-1).
%F a(n) = T(n, 6) = (S(n, 12)-S(n-2, 12))/2 with S(n, x) := U(n, x/2) and T(n), resp. U(n, x), are Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(-2, x) := -1, S(-1, x) := 0, S(n, 12)=A004191(n).
%F a(n) = ((6+sqrt(35))^n + (6-sqrt(35))^n)/2.
%F G.f.: (1-6*x)/(1-12*x+x^2).
%F a(n)*a(n+3) - a(n+1)*a(n+2) = 420. - _Ralf Stephan_, Jun 06 2005
%p A023038:=n->round(((6+sqrt(35))^n + (6-sqrt(35))^n)/2); seq(A023038(n), n=0..30); # _Wesley Ivan Hurt_, Feb 03 2014
%t Table[Round[((6 + Sqrt[35])^n + (6 - Sqrt[35])^n)/2], {n, 0, 30}] (* _Wesley Ivan Hurt_, Feb 03 2014 *)
%t nn = 20; CoefficientList[Series[(1 - 6*x)/(1 - 12*x + x^2), {x, 0, nn}], x] (* _T. D. Noe_, Feb 05 2014 *)
%o (PARI) x='x+O('x^30); Vec((1-6*x)/(1-12*x+x^2)) \\ _G. C. Greubel_, Dec 19 2017
%o (Magma) I:=[1, 6]; [n le 2 select I[n] else 12*Self(n-1) - Self(n-2): n in [1..30]]; // _G. C. Greubel_, Dec 19 2017
%Y Cf. A087800.
%K nonn,easy
%O 0,2
%A _David W. Wilson_
| 