A195623 Numerators of Pythagorean approximations to 5.

99, 10101, 1030199, 105070201, 10716130299, 1092940220301, 111469186340399, 11368764066500401, 1159502465596700499, 118257882726796950501, 12061144535667692250599, 1230118484755377812610601, 125460024300512869194030699, 12795692360167557279978520701, 1305035160712790329688615080799

Offset: 1

Author

Clark Kimberling, Sep 22 2011

Keywords

• nonn
• easy
• frac

Comments

See A195500 for discussion and list of related sequences; see A195622 for Mathematica program.

Links

• Colin Barker, Table of n, a(n) for n = 1..497
• Index entries for linear recurrences with constant coefficients, signature (101,101,-1).

Crossrefs

Cf. A097726, A097727, A195500, A195622.

Programs

Magma

I:=[99,10101,1030199]; [n le 3 select I[n] else 101*Self(n-1) +101*Self(n-2) -Self(n-3): n in [1..40]]; // G. C. Greubel, Feb 16 2023

Mathematica

Table[(5*LucasL[2*n+1,10] +2*(-1)^n)/52, {n,40}] (* G. C. Greubel, Feb 16 2023 *)

PARI

Vec(-x*(x^2-102*x-99) / ((x+1)*(x^2-102*x+1)) + O(x^20)) \\ Colin Barker, Jun 03 2015

SageMath

A097726=BinaryRecurrenceSequence(102, -1, 1, 103)
[(1/26)*(25*A097726(n) + (-1)^n) for n in range(1, 41)] # G. C. Greubel, Feb 16 2023

Formula

From Colin Barker, Jun 03 2015: (Start)

a(n) = 101*a(n-1) + 101*a(n-2) - a(n-3).

G.f.: x*(99+102*x-x^2)/((1+x)*(1-102*x+x^2)). (End)

a(n) = (1/26)*(25*A097726(n) + (-1)^n). - G. C. Greubel, Feb 16 2023

E.g.f.: (5*exp(51*x)*(5*cosh(10*sqrt(26)*x) + sqrt(26)*sinh(10*sqrt(26)*x)) + exp(-x) - 26)/26. - Stefano Spezia, Aug 05 2024