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A136211 Denominators in continued fraction [0; 1, 3, 1, 3, 1, 3,...]. 5
1, 4, 5, 19, 24, 91, 115, 436, 551, 2089, 2640, 10009, 12649, 47956, 60605, 229771, 290376, 1100899, 1391275, 5274724, 6665999, 25272721, 31938720, 121088881, 153027601, 580171684, 733199285, 2779769539, 3512968824 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A136210(n)/A136211(n) tends to .791287847... = [0; 1, 3, 1, 3, 1, 3,...] = (sqrt(21) - 3)/2 = the inradius of a right triangle with hypotenuse 3, legs 2 and sqrt(21).

The number .791287847... = (sqrt(21) - 3)/2 arises in finding a number which is 5 less than its square; the result is: 2.791287847... because (2.791287847...)^2 = 7.791287847... In general the quadratic equation for finding such numbers is x^2 - x = N, so x = (1 + sqrt(1 + 4N))/2. - Alexander R. Povolotsky, Dec 23 2007

Prepending a 1 to the sequence gives [1, 1, 4, 5, 19, 24,...]. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) with the parameters R = 3 and Q = -1. It is a strong divisibility sequence, that is, GCD(a(n),a(m)) = a(GCD(n,m)) for all natural numbers n and m. - Peter Bala, May 14 2014

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

P. Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6

Eric Weisstein's World of Mathematics, Lehmer Number

Index entries for linear recurrences with constant coefficients, signature (0, 5, 0, -1).

FORMULA

a(1) = 1, a(2) = 4, then for n>2, a(2n) = 3*a(2n-1) + a(2n-2); a(2n-1) = a(2n-2) + a(2n-3). Let T = the 2 X 2 matrix [1, 3; 1, 4]. Then T^n = [A136210(2n-1), A136210(2n); a(2n-1), a(2n)].

From R. J. Mathar, May 18 2008: (Start)

O.g.f.: x*(1+4*x-x^3)/(1-5*x^2+x^4).

a(2*n) = A004253(n+1).

a(2*n+1) = A004254(n). (End)

a(n)*a(n+1) = A099025(n). - R. K. Guy, May 18 2008

{-a(n) + 5 a(n + 2) - a(n + 4), a(0) = 1, a(1) = 4, a(2) = 5, a(3) = 19}. - Robert Israel, May 14 2008

EXAMPLE

a(4) = 19 = 3*a(3) + a(2) = 3*5 + 4.

a(5) = 24 = a(4) + a(3) = 19 + 5.

T^3 = [19, 72; 24, 91], where the bottom row [24, 91] = [a(5), a(6)].

MATHEMATICA

Denominator[NestList[(3/(3+#))&, 0, 60]] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2010 *)

a[n_] := FromContinuedFraction[ Join[{0}, 3 - 2*Array[Mod[#, 2]&, n]]] // Denominator; Table[a[n], {n, 1, 30}] (* Jean-Fran├žois Alcover, May 15 2014 *)

PROG

(PARI) x='x + O('x^25); Vec(x*(1+4*x-x^3)/(1-5*x^2+x^4)) \\ G. C. Greubel, Feb 18 2017

CROSSREFS

Cf. A136210.

Sequence in context: A217686 A042885 A042085 * A041036 A041699 A154704

Adjacent sequences:  A136208 A136209 A136210 * A136212 A136213 A136214

KEYWORD

nonn,frac

AUTHOR

Gary W. Adamson, Dec 21 2007

STATUS

approved

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Last modified October 22 12:47 EDT 2019. Contains 328318 sequences. (Running on oeis4.)