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%I #53 Aug 24 2020 23:14:47
%S 2,3,14,17,82,99,478,577,2786,3363,16238,19601,94642,114243,551614,
%T 665857,3215042,3880899,18738638,22619537,109216786,131836323,
%U 636562078,768398401,3710155682,4478554083,21624372014,26102926097,126036076402,152139002499
%N Numerators of continued fraction convergents to sqrt(8).
%H Vincenzo Librandi, <a href="/A041010/b041010.txt">Table of n, a(n) for n = 0..199</a> [1 removed by _Georg Fischer_, Jul 01 2019]
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,0,-1).
%F a(n) = 6*a(n-2) - a(n-4).
%F a(2n) = a(2n-1) + a(2n-2), a(2n+1) = 4*a(2n) + a(2n-1).
%F a(2n) = A001333(2n), a(2n+1) = 2*A001333(2n+1).
%F G.f.: (2+3*x+2*x^2-x^3)/(1-6*x^2+x^4).
%F a(n) = A001333(n+1)*A000034(n+1). - _R. J. Mathar_, Jul 08 2009
%F From _Gerry Martens_, Jul 11 2015: (Start)
%F Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
%F a0(n) = -((3-2*sqrt(2))^n*(1+sqrt(2))) + (-1+sqrt(2))*(3+2*sqrt(2))^n.
%F a1(n) = ((3-2*sqrt(2))^n + (3+2*sqrt(2))^n)/2. (End)
%t Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[8],n]]],{n,1,50}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 16 2011*)
%t CoefficientList[Series[(2 + 3*x + 2*x^2 - x^3)/(1 - 6*x^2 + x^4), {x, 0, 30}], x] (* _Vincenzo Librandi_, Oct 28 2013 *)
%t a0[n_] := -((3-2*Sqrt[2])^n*(1+Sqrt[2]))+(-1+Sqrt[2])*(3+2*Sqrt[2])^n // Simplify
%t a1[n_] := ((3-2*Sqrt[2])^n+(3+2*Sqrt[2])^n)/2 // Simplify
%t Flatten[MapIndexed[{a0[#], a1[#]} &,Range[20]]] (* _Gerry Martens_, Jul 11 2015 *)
%o From _M. F. Hasler_, Nov 01 2019: (Start)
%o (PARI) A041010=contfracpnqn(c=contfrac(sqrt(8)),#c)[1,][^-1] \\ Discard possibly incorrect last element. NB: a(n)=A041010[n+1]! For more terms use:
%o A041010(n)={n<#A041010|| A041010=extend(A041010, [-1,0,6,0]~, n\.8); A041010[n+1]}
%o extend(A,c,N)={for(n=#A+1,#A=Vec(A,N), A[n]=A[n-#c..n-1]*c);A} \\ (End)
%Y Cf. A040005 (continued fraction), A041011 (denominators), A010466 (decimals).
%Y Analog for other sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041006 (m=6), A041008 (m=7), A005667 (m=10), A041014 (m=11), A041016 (m=12), ..., A042934 (m=999), A042936 (m=1000).
%K nonn,cofr,frac,easy
%O 0,1
%A _N. J. A. Sloane_
%E Entry improved by _Michael Somos_
%E Initial term 1 removed and b-file, program and formulas adapted by _Georg Fischer_, Jul 01 2019
%E Cross-references added by _M. F. Hasler_, Nov 02 2019
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