A195619 Denominators of Pythagorean approximations to 4.

16, 1040, 68640, 4529184, 298857520, 19720067120, 1301225572416, 85861167712320, 5665535843440720, 373839504499375184, 24667741761115321440, 1627697116729111839840, 107403341962360266108016, 7086992872399048451289200

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..549
• Index entries for linear recurrences with constant coefficients, signature (65,65,-1).

Crossrefs

Cf. A078988, A078989, A195500, A195620.

Programs

Magma

I:=[16, 1040, 68640]; [n le 3 select I[n] else 65*Self(n-1) +65*Self(n-2) -Self(n-3): n in [1..40]]; // G. C. Greubel, Feb 13 2023

Mathematica

r = 4; 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}]]  (* A195619, A195620 *)
Sqrt[a^2 + b^2] (* A078988 *)
(* Peter J. C. Moses, Sep 02 2011 *)
Table[(LucasL[2*n+1,8] - 8*(-1)^n)/34, {n,40}] (* G. C. Greubel, Feb 13 2023 *)
LinearRecurrence[{65,65,-1},{16,1040,68640},20] (* Harvey P. Dale, May 01 2023 *)

PARI

Vec(16*x/((x+1)*(x^2-66*x+1)) + O(x^20)) \\ Colin Barker, Jun 03 2015

SageMath

A078989=BinaryRecurrenceSequence(66,-1,1,67)
[4*(A078989(n) - (-1)^n)/17 for n in range(1,41)] # G. C. Greubel, Feb 13 2023

Formula

From Colin Barker, Jun 03 2015: (Start)

a(n) = 65*a(n-1) + 65*a(n-2) - a(n-3).

G.f.: 16*x/((1+x)*(1-66*x+x^2)). (End)

a(n) = ((4+sqrt(17))^(2*n+1) + (4-sqrt(17))^(2*n+1) - 8*(-1)^n)/34. - Colin Barker, Mar 03 2016

a(n) = 4*(A078989(n) - (-1)^n)/17. - G. C. Greubel, Feb 13 2023