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 A195614 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 2. 3
 8, 136, 2448, 43920, 788120, 14142232, 253772064, 4553754912, 81713816360, 1466294939560, 26311595095728, 472142416783536, 8472251907007928, 152028391909359160, 2728038802461456960, 48952670052396866112, 878420022140682133064 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS See A195500 for a discussion and references. LINKS Colin Barker, Table of n, a(n) for n = 1..797 Index entries for linear recurrences with constant coefficients, signature (17,17,-1). FORMULA From Colin Barker, Jun 04 2015: (Start) G.f.: 8*x / ((x+1)*(x^2-18*x+1)). a(n) = 17*a(n-1) + 17*a(n-2) - a(n-3). (End) a(n) = (-4*(-1)^n - (-2+sqrt(5))*(9+4*sqrt(5))^(-n) + (2+sqrt(5))*(9+4*sqrt(5))^n)/10. - Colin Barker, Mar 04 2016 a(n) = A014445(n) * A014445(n+1) / 2. - Diego Rattaggi, Jun 01 2020 a(n) is the numerator of continued fraction [4, ..., 4, 8, 4, ..., 4] with (n-1) 4's before and after the middle 8. - Greg Dresden and Hexuan Wang, Aug 30 2021 MATHEMATICA r = 2; z = 32; p[{f_, n_}] := (#1[[2]]/#1[[       1]] &)[({2 #1[[1]] #1[[2]], #1[[1]]^2 - #1[[          2]]^2} &)[({Numerator[#1], Denominator[#1]} &)[      Array[FromContinuedFraction[         ContinuedFraction[(#1 + Sqrt[1 + #1^2] &)[f], #1]] &, {n}]]]]; {a, b} = ({Denominator[#1], Numerator[#1]} &)[   p[{r, z}]]  (* A195614, A195615 *) Sqrt[a^2 + b^2] (* A007805 *) (* Peter J. C. Moses, Sep 02 2011 *) PROG (PARI) Vec(8*x/((x+1)*(x^2-18*x+1)) + O(x^50)) \\ Colin Barker, Jun 04 2015 CROSSREFS Cf. A007805, A014445, A195500, A195615. Sequence in context: A291699 A292914 A072072 * A131927 A132869 A036915 Adjacent sequences:  A195611 A195612 A195613 * A195615 A195616 A195617 KEYWORD nonn,easy AUTHOR Clark Kimberling, Sep 22 2011 STATUS approved

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Last modified August 16 01:49 EDT 2022. Contains 356150 sequences. (Running on oeis4.)