A083043 - OEIS

%I #29 Sep 08 2022 08:45:10

%S 1,21,419,8359,166761,3326861,66370459,1324082319,26415275921,

%T 526981436101,10513213446099,209737287485879,4184232536271481,

%U 83474913437943741,1665314036222603339,33222805811014123039

%N Integers y such that 11*x^2 - 9*y^2 = 2 for some integer x.

%H G. C. Greubel, <a href="/A083043/b083043.txt">Table of n, a(n) for n = 1..750</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (20,-1).

%F G.f.: x*(1+x)/(1-20*x+x^2).

%F a(n) = 20*a(n-1) - a(n-2).

%F a(1-n) = -a(n).

%F 11*A075839(n)^2 - 9*a(n)^2 = 2.

%F a(n+1) = 10*a(n) + sqrt(99*a(n)^2 + 22). - _Richard Choulet_, Sep 27 2007

%F a(n) = ((3 + sqrt(11))*(10 + 3*sqrt(11))^(n-1) + (3 - sqrt(11))*(10 - 3*sqrt(11))^(n-1))/6. - _G. C. Greubel_, Dec 06 2019

%F E.g.f.: 1 + (1/3)*exp(10*x)*(-3*cosh(3*sqrt(11)*x) + sqrt(11)*sinh(3*sqrt(11)*x)). - _Stefano Spezia_, Dec 06 2019 after _G. C. Greubel_

%p seq(coeff(series( x*(1+x)/(1-20*x+x^2), x, n+1), x, n), n = 1..20); # _G. C. Greubel_, Dec 06 2019

%t LinearRecurrence[{20,-1},{1,21},20] (* _Harvey P. Dale_, Jun 02 2014 *)

%o (PARI) a(n)=subst(poltchebi(n+1)-poltchebi(n),x,10)/9

%o (Magma) I:=[1,21]; [n le 2 select I[n] else 20*Self(n-1) - Self(n-2): n in [1..20]]; // _G. C. Greubel_, Dec 06 2019

%o (Sage)

%o def A083043_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( x*(1+x)/(1-20*x+x^2) ).list()

%o a=A083043_list(20); a[1:] # _G. C. Greubel_, Dec 06 2019

%o (GAP) a:=[1,21];; for n in [3..20] do a[n]:=20*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Dec 06 2019

%Y Cf. A075839, A075843.

%K easy,nonn

%O 1,2

%A _Michael Somos_, Apr 17 2003

%E Corrected by _T. D. Noe_, Nov 07 2006

%E Offset changed to 1 by _G. C. Greubel_, Dec 06 2019