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A163271 Numerators of fractions in a 'zero-transform' approximation of sqrt(2) by means of a(n) = (a(n-1) + c)/(a(n-1) + 1) with c=2 and a(1)=0.

%I A163271 #71 Sep 11 2020 22:26:53
%S A163271 0,2,4,10,24,58,140,338,816,1970,4756,11482,27720,66922,161564,390050,
%T A163271 941664,2273378,5488420,13250218,31988856,77227930,186444716,
%U A163271 450117362,1086679440,2623476242,6333631924,15290740090,36915112104,89120964298,215157040700
%N A163271 Numerators of fractions in a 'zero-transform' approximation of sqrt(2) by means of a(n) = (a(n-1) + c)/(a(n-1) + 1) with c=2 and a(1)=0.
%C A163271 Twice the Pell numbers; for denominators see A001333 (numerators of the approximation of sqrt(2) for a(1) = 1).
%C A163271 Row sums of the triangle A128966. - _Reinhard Zumkeller_, Jul 20 2013
%C A163271 Because a(n+1)/A001333(n) approximates sqrt(2) and a(n) = 2*A001333(n) - a(n+1), we get that a(n)/A001333(n) approximates 2 - sqrt(2). - _Danny Rorabaugh_, Mar 14 2015
%C A163271 Number of weak orderings R on {1,...,n} that are weakly single-peaked w.r.t. the total ordering 1 < ... < n and for which {1,...,n} has exactly one minimal and one maximal element for the weak ordering R. - _J. Devillet_, Sep 28 2017
%H A163271 Reinhard Zumkeller, <a href="/A163271/b163271.txt">Table of n, a(n) for n = 1..1000</a>
%H A163271 M. Couceiro, J. Devillet, and J.-L. Marichal, <a href="http://arxiv.org/abs/1709.09162">Quasitrivial semigroups: characterizations and enumerations</a>, arXiv:1709.09162 [math.RA],  2017.
%H A163271 I. M. Gessel, Ji Li, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Gessel/gessel6.html">Compositions and Fibonacci identities</a>, J. Int. Seq. 16 (2013) 13.4.5.
%H A163271 Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, Minghao Chen, <a href="https://doi.org/10.3934/era.2020057">Recursive sequences and Girard-Waring identities with applications in sequence transformation</a>, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.
%H A163271 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,1).
%F A163271 a(n) = A052542(n-1), n > 1.
%F A163271 G.f.: x + x^2/(2*G(0)-x) where G(k) = 1 - (k+1)/(1 - x/(x +(k+1)/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Dec 07 2012
%F A163271 a(n) = 2*a(n-1) + a(n-2). - _Jacob Antony_, Jun 07 2013
%F A163271 a(n) = b(n) - b(n-1) = 2b(n-1) - a(n-1) where b(n) = A001333(n). - _Danny Rorabaugh_, Mar 14 2015
%F A163271 G.f.: 2*t^2/(1 - 2*t - t^2). - _G. C. Greubel_, Dec 12 2016
%F A163271 a(n) = 2*A000129(n-1) (see the first comment). - _J. Devillet_, Sep 28 2017
%p A163271 A163271:=gfun:-rectoproc({a(n) = 2 * a(n-1) + a(n-2), a(1) = 0, a(2) = 2}, a(n), remember):  map(A163271, [$1..100]);  # _Muniru A Asiru_, Oct 08 2017
%t A163271 CoefficientList[Series[2*t^2/(1-2*t - t^2), {t,0,50}], t] (* or *) LinearRecurrence[{2,1},{0,2},50] (* _G. C. Greubel_, Dec 12 2016 *)
%o A163271 (Haskell)
%o A163271 a163271 = sum . a128966_row . (subtract 1)
%o A163271 -- _Reinhard Zumkeller_, Jul 20 2013
%o A163271 (PARI) concat([0], Vec(2*t^2/(1-2*t - t^2) + O(t^50))) \\ _G. C. Greubel_, Dec 12 2016
%o A163271 (GAP)
%o A163271 a := [0, 2];; for n in [3..10^2] do a[n] := 2*a[n-1] + a[n-2]; od; A163271:=a; # _Muniru A Asiru_, Oct 08 2017
%Y A163271 Cf. A000129 (Pell numbers), A001333 (denominators), A052542.
%Y A163271 Cf. A014113, A189687.
%Y A163271 Cf. A293004.
%K A163271 easy,nonn
%O A163271 1,2
%A A163271 _Mark Dols_, Jul 24 2009