A195556 - OEIS

A195556

Denominators a(n) of Pythagorean approximations b(n)/a(n) to 1/3.

1, 12, 24, 35, 468, 900, 1333, 17760, 34188, 50615, 674424, 1298232, 1922041, 25610340, 49298640, 72986939, 972518508, 1872050076, 2771581645, 36930092952, 71088604260, 105247115567, 1402371013680, 2699494911792, 3996618809905

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OFFSET

1,2

COMMENTS

See A195500 for a discussion and references.

LINKS

Table of n, a(n) for n=1..25.

FORMULA

Conjecture: a(n) = 37*a(n-3) + 37*a(n-6) - a(n-9). - R. J. Mathar, Sep 21 2011

Empirical g.f.: x*(x^6+12*x^5+24*x^4-2*x^3+24*x^2+12*x+1) / (x^9-37*x^6-37*x^3+1). - Colin Barker, Jun 04 2015

MATHEMATICA

r = 1/3; z = 27;

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}]] (* A195556, A195557 *)

Sqrt[a^2 + b^2] (* A195558 *)

(* Peter J. C. Moses, Sep 02 2011 *)

CROSSREFS

Cf. A195500, A195557, A195558.

Sequence in context: A336772 A098113 A172415 * A083547 A009185 A102308

Adjacent sequences: A195553 A195554 A195555 * A195557 A195558 A195559

KEYWORD

nonn

AUTHOR

Clark Kimberling, Sep 21 2011

STATUS

approved