A195616 Denominators of Pythagorean approximations to 3.

12, 444, 16872, 640680, 24328980, 923860548, 35082371856, 1332206269968, 50588755886940, 1921040517433740, 72948950906595192, 2770139093933183544, 105192336618554379492, 3994538652411133237140, 151687276455004508631840

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..633
• Index entries for linear recurrences with constant coefficients, signature (37,37,-1).

Crossrefs

Cf. A085447, A097314, A097315, A195500, A195617.

Programs

Magma

I:=[12, 444, 16872]; [n le 3 select I[n] else 37*Self(n-1) +37*Self(n-2) -Self(n-3): n in [1..40]]; // G. C. Greubel, Feb 13 2023

Mathematica

r = 3; 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}]]  (* A195616, A195617 *)
Sqrt[a^2 + b^2] (* A097315 *)
(* Peter J. C. Moses, Sep 02 2011 *)
Table[(1/20)*(LucasL[2*n+1,6] -6*(-1)^n), {n,40}] (* G. C. Greubel, Feb 13 2023 *)

PARI

Vec(12*x/((1+x)*(1-38*x+x^2)) + O(x^20)) \\ Colin Barker, Jun 04 2015

SageMath

A085447=BinaryRecurrenceSequence(6,1,2,6)
[(A085447(2*n+1) - 6*(-1)^n)/20 for n in range(1,41)] # G. C. Greubel, Feb 13 2023

Formula

From Colin Barker, Jun 04 2015: (Start)

a(n) = 37*a(n-1) + 37*a(n-2) - a(n-3).

G.f.: 12*x / ((1+x)*(1-38*x+x^2)). (End)

From G. C. Greubel, Feb 13 2023: (Start)

a(n) = (3/10)*(A097314(n) + (-1)^n).

a(n) = (1/20)*(A085447(2*n+1) - 6*(-1)^n). (End)