A195620 Numerators of Pythagorean approximations to 4.

63, 4161, 274559, 18116737, 1195430079, 78880268481, 5204902289663, 343444670849281, 22662143373762879, 1495358017997500737, 98670967044461285759, 6510788466916447359361, 429613367849441064432063, 28347971489596193805156801

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

Crossrefs

Cf. A078989, A195500, A195619, A195620.

Programs

Magma

I:=[63,4161,274559]; [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 15 2023

Mathematica

LinearRecurrence[{65,65,-1}, {63,4161,274559}, 40] (* G. C. Greubel, Feb 15 2023 *)

PARI

Vec(x*(63+66*x-x^2)/((1+x)*(1-66*x+x^2)) + O(x^20)) \\ Colin Barker, Jun 03 2015

SageMath

A078989=BinaryRecurrenceSequence(66, -1, 1, 67)
[(16*A078989(n) + (-1)^n)/17 for n in range(1, 41)] # G. C. Greubel, Feb 15 2023

Formula

From Colin Barker, Jun 03 2015: (Start)

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

G.f.: x*(63+66*x-x^2) / ((1+x)*(1-66*x+x^2)). (End)

a(n) = ((-1)^n - 2*(-4+sqrt(17))*(33+8*sqrt(17))^(-n) + 2*(4+sqrt(17))*(33+8*sqrt(17))^n)/17. - Colin Barker, Mar 03 2016

a(n) = (1/17)*(A078989(n) + (-1)^n) - [n=0]. - G. C. Greubel, Feb 15 2023