A195622 Denominators of Pythagorean approximations to 5.

20, 2020, 206040, 21014040, 2143226060, 218588044060, 22293837268080, 2273752813300080, 231900493119340100, 23651576545359390100, 2412228907133538450120, 246023696951075562522120, 25092004860102573838806140, 2559138472033511455995704140

Offset: 1

Author

Clark Kimberling, Sep 22 2011

Keywords

• nonn
• easy
• frac

Comments

See A195500 for a discussion and references.

Links

• Colin Barker, Table of n, a(n) for n = 1..498
• Index entries for linear recurrences with constant coefficients, signature (101,101,-1).

Crossrefs

Cf. A097726, A195500, A195623.

Programs

Magma

I:=[20,2020,206040]; [n le 3 select I[n] else 101*Self(n-1) +101*Self(n-2) -Self(n-3): n in [1..40]]; // G. C. Greubel, Feb 15 2023

Mathematica

r = 5; z = 20;
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}]]  (* A195622, A195623 *)
Sqrt[a^2 + b^2] (* A097727 *)
(* by Peter J. C. Moses, Sep 02 2011 *)
LinearRecurrence[{101,101,-1},{20,2020,206040},20] (* Harvey P. Dale, Oct 17 2021 *)

PARI

Vec(20*x/((x+1)*(x^2-102*x+1)) + O(x^20)) \\ Colin Barker, Jun 03 2015

SageMath

A097726=BinaryRecurrenceSequence(102, -1, 1, 103)
[(5/26)*(A097726(n) - (-1)^n) for n in range(1, 41)] # G. C. Greubel, Feb 15 2023

Formula

From Colin Barker, Jun 03 2015: (Start)

a(n) = 101*a(n-1) + 101*a(n-2) - a(n-3).

G.f.: 20*x/((1+x)*(1-102*x+x^2)). (End)

a(n) = (5/26)*(A097726(n) - (-1)^n). - G. C. Greubel, Feb 15 2023