login
A195619
Denominators of Pythagorean approximations to 4.
4
16, 1040, 68640, 4529184, 298857520, 19720067120, 1301225572416, 85861167712320, 5665535843440720, 373839504499375184, 24667741761115321440, 1627697116729111839840, 107403341962360266108016, 7086992872399048451289200
OFFSET
1,1
COMMENTS
See A195500 for a discussion and references.
FORMULA
From Colin Barker, Jun 03 2015: (Start)
a(n) = 65*a(n-1) + 65*a(n-2) - a(n-3).
G.f.: 16*x/((1+x)*(1-66*x+x^2)). (End)
a(n) = ((4+sqrt(17))^(2*n+1) + (4-sqrt(17))^(2*n+1) - 8*(-1)^n)/34. - Colin Barker, Mar 03 2016
a(n) = 4*(A078989(n) - (-1)^n)/17. - G. C. Greubel, Feb 13 2023
MATHEMATICA
r = 4; z = 20;
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}]] (* A195619, A195620 *)
Sqrt[a^2 + b^2] (* A078988 *)
(* Peter J. C. Moses, Sep 02 2011 *)
Table[(LucasL[2*n+1, 8] - 8*(-1)^n)/34, {n, 40}] (* G. C. Greubel, Feb 13 2023 *)
LinearRecurrence[{65, 65, -1}, {16, 1040, 68640}, 20] (* Harvey P. Dale, May 01 2023 *)
PROG
(PARI) Vec(16*x/((x+1)*(x^2-66*x+1)) + O(x^20)) \\ Colin Barker, Jun 03 2015
(Magma) I:=[16, 1040, 68640]; [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 13 2023
(SageMath)
A078989=BinaryRecurrenceSequence(66, -1, 1, 67)
[4*(A078989(n) - (-1)^n)/17 for n in range(1, 41)] # G. C. Greubel, Feb 13 2023
CROSSREFS
KEYWORD
nonn,easy,frac
AUTHOR
Clark Kimberling, Sep 22 2011
STATUS
approved