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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. 11
 0, 2, 4, 10, 24, 58, 140, 338, 816, 1970, 4756, 11482, 27720, 66922, 161564, 390050, 941664, 2273378, 5488420, 13250218, 31988856, 77227930, 186444716, 450117362, 1086679440, 2623476242, 6333631924, 15290740090, 36915112104, 89120964298, 215157040700 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Twice the Pell numbers; for denominators see A001333 (numerators of the approximation of sqrt(2) for a(1) = 1). Row sums of the triangle A128966. - Reinhard Zumkeller, Jul 20 2013 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 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 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..1000 M. Couceiro, J. Devillet, and J.-L. Marichal, Quasitrivial semigroups: characterizations and enumerations, arXiv:1709.09162 [math.RA], 2017. I. M. Gessel, Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5. Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, Minghao Chen, Recursive sequences and Girard-Waring identities with applications in sequence transformation, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062. Index entries for linear recurrences with constant coefficients, signature (2,1). FORMULA a(n) = A052542(n-1), n > 1. 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 a(n) = 2*a(n-1) + a(n-2). - Jacob Antony, Jun 07 2013 a(n) = b(n) - b(n-1) = 2b(n-1) - a(n-1) where b(n) = A001333(n). - Danny Rorabaugh, Mar 14 2015 G.f.: 2*t^2/(1 - 2*t - t^2). - G. C. Greubel, Dec 12 2016 a(n) = 2*A000129(n-1) (see the first comment). - J. Devillet, Sep 28 2017 MAPLE 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 MATHEMATICA 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 *) PROG (Haskell) a163271 = sum . a128966_row . (subtract 1) -- Reinhard Zumkeller, Jul 20 2013 (PARI) concat([0], Vec(2*t^2/(1-2*t - t^2) + O(t^50))) \\ G. C. Greubel, Dec 12 2016 (GAP) 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 CROSSREFS Cf. A000129 (Pell numbers), A001333 (denominators), A052542. Cf. A014113, A189687. Cf. A293004. Sequence in context: A025275 A165409 A052542 * A178036 A191625 A253014 Adjacent sequences: A163268 A163269 A163270 * A163272 A163273 A163274 KEYWORD easy,nonn AUTHOR Mark Dols, Jul 24 2009 STATUS approved

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Last modified August 8 12:42 EDT 2024. Contains 375021 sequences. (Running on oeis4.)