A195620 Numerators of Pythagorean approximations to 4.

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

Offset

1,1

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).

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

Mathematica

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

Prog

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

Crossrefs

Cf. A078989, A195500, A195619, A195620.

Sequence in context: A267963 A268028 A194484 * A238994 A364745 A069407

Adjacent sequences: A195617 A195618 A195619 * A195621 A195622 A195623

Keyword

nonn,easy

Author

Clark Kimberling, Sep 22 2011

Status

approved