A145542 Numerators in continued fraction expansion of sqrt(3/5).

1, 3, 7, 24, 55, 189, 433, 1488, 3409, 11715, 26839, 92232, 211303, 726141, 1663585, 5716896, 13097377, 45009027, 103115431, 354355320, 811826071, 2789833533, 6391493137, 21964312944, 50320119025, 172924670019, 396169459063, 1361433047208, 3119035553479

Offset: 1

Author

Gary W. Adamson, Oct 12 2008

Keywords

• nonn
• easy

Comments

a(n)/A145543(n) tends to sqrt(3/5).

A strong divisibility sequence, that is gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Related to the Lehmer sequence U_n(sqrt(R),Q) with parameters R = 6 and Q = -1. See A041023. - Peter Bala, Jun 06 2014

Examples

			[a(7), a(8)] = [433, 1488] X^4 * [1, 0] = [433, 1488].
a(5) = 55 = 2*a(4) + a(3) = 2*24 + 7.
G.f. = x + 3*x^2 + 7*x^3 + 24*x^4 + 55*x^5 + 189*x^6 + 433*x^7 + 1488*x^8 + ...
		

Links

• Peter Bala, Notes on 2-periodic continued fractions and Lehmer sequences
• Eric Weisstein's World of Mathematics, Lehmer Number

Crossrefs

Cf. A001090, A041203, A070997, A145543.

Programs

Mathematica

Numerator[Convergents[Sqrt[3/5], 30]] (* gives terms with 0 prepended *) (* Wesley Ivan Hurt, Jun 15 2014 *)

PARI

{a(n) = if( n<1, 0, polcoeff( x * (1 + 3*x - x^2) / (1 - 8*x^2 + x^4) + x * O(x^n), n))}; /* Michael Somos, Nov 14 2015 */

Formula

Numerators in continued fraction expansion of sqrt(3/5); i.e., of [1, 3, 2, 3, 2, 3, 2, 3, 2, ...].

[a(2*n - 1), a(2*n)] = X^n * [1,0], where X is the 2 X 2 matrix [1,2; 3,7].

Empirical G.f.: x*(1+3*x-x^2)/(1-8*x^2+x^4). - Colin Barker, Jan 04 2012

From Peter Bala, Jun 06 2014: (Start)

a(2*n + 1) = Product_{k=1..n} (6 + 4*cos^2(k*Pi/(2*n+1))).

a(2*n) = 3*Product_{k=1..n-1} (6 + 4*cos^2(k*Pi/(2*n))).

a(2*n + 1) = A070997(n); a(2*n) = 3*A001090(n). (End)

Extensions

More terms from Wesley Ivan Hurt, Jun 15 2014