A195617 Numerators b(n) of Pythagorean approximations b(n)/a(n) to 3.

35, 1333, 50615, 1922041, 72986939, 2771581645, 105247115567, 3996618809905, 151766267660819, 5763121552301221, 218846852719785575, 8310417281799550633, 315577009855663138475, 11983615957233399711421, 455061829365013525895519

Offset: 1

Author

Clark Kimberling, Sep 22 2011

Keywords

• nonn
• easy

Comments

See A195500 for discussion and list of related sequences; see A195616 for Mathematica program.

Links

• Colin Barker, Table of n, a(n) for n = 1..632
• Index entries for linear recurrences with constant coefficients, signature (37,37,-1).

Crossrefs

Cf. A085447, A097315, A195500, A195616.

Programs

Magma

I:=[35, 1333, 50615]; [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

Table[(3*LucasL[2*n+1,6] +2*(-1)^n)/20, {n, 40}] (* G. C. Greubel, Feb 13 2023 *)

PARI

Vec(-x*(x^2-38*x-35)/((x+1)*(x^2-38*x+1)) + O(x^50)) \\ Colin Barker, Jun 04 2015

SageMath

A085447=BinaryRecurrenceSequence(6,1,2,6)
[(3*A085447(2*n+1) + 2*(-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.: x*(35+38*x-x^2) / ((1+x)*(1-38*x+x^2)). (End)

a(n) = (1/20)*(3*A085447(2*n+1) + 2*(-1)^n). - G. C. Greubel, Feb 13 2023