A195614 - OEIS

%I #41 Oct 13 2021 21:03:29

%S 8,136,2448,43920,788120,14142232,253772064,4553754912,81713816360,

%T 1466294939560,26311595095728,472142416783536,8472251907007928,

%U 152028391909359160,2728038802461456960,48952670052396866112,878420022140682133064

%N Denominators a(n) of Pythagorean approximations b(n)/a(n) to 2.

%C See A195500 for a discussion and references.

%H Colin Barker, <a href="/A195614/b195614.txt">Table of n, a(n) for n = 1..797</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (17,17,-1).

%F From _Colin Barker_, Jun 04 2015: (Start)

%F G.f.: 8*x / ((x+1)*(x^2-18*x+1)).

%F a(n) = 17*a(n-1) + 17*a(n-2) - a(n-3). (End)

%F 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

%F a(n) = A014445(n) * A014445(n+1) / 2. - _Diego Rattaggi_, Jun 01 2020

%F 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

%t r = 2; z = 32;

%t p[{f_, n_}] := (#1[[2]]/#1[[

%t       1]] &)[({2 #1[[1]] #1[[2]], #1[[1]]^2 - #1[[

%t          2]]^2} &)[({Numerator[#1], Denominator[#1]} &)[

%t      Array[FromContinuedFraction[

%t         ContinuedFraction[(#1 + Sqrt[1 + #1^2] &)[f], #1]] &, {n}]]]];

%t {a, b} = ({Denominator[#1], Numerator[#1]} &)[

%t   p[{r, z}]]  (* A195614, A195615 *)

%t Sqrt[a^2 + b^2] (* A007805 *)

%t (* _Peter J. C. Moses_, Sep 02 2011 *)

%o (PARI) Vec(8*x/((x+1)*(x^2-18*x+1)) + O(x^50)) \\ _Colin Barker_, Jun 04 2015

%Y Cf. A007805, A014445, A195500, A195615.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Sep 22 2011