A112447 a(2*n) = A001045(n+2); a(2*n+1) = A001045(n+1).

1, 1, 3, 1, 5, 3, 11, 5, 21, 11, 43, 21, 85, 43, 171, 85, 341, 171, 683, 341, 1365, 683, 2731, 1365, 5461, 2731, 10923, 5461, 21845, 10923, 43691, 21845, 87381, 43691, 174763, 87381, 349525, 174763, 699051, 349525, 1398101, 699051, 2796203, 1398101, 5592405

Offset: 0

Author

Edwin F. Sampang, Dec 12 2005

Keywords

• nonn
• easy

Comments

Consider the Harmonacci sequence: H(1)=x, H(2)=y, H(3)=2xy/(x+y), H(4)=4xy/(3x+y)...; H(m) is the harmonic mean of H(m-1) and H(m-2). a(2n) and a(2n+1) are the denominator coefficients of H(n+3).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,1,0,2).

Crossrefs

Cf. A001045.

Programs

Mathematica

LinearRecurrence[{0,1,0,2},{1,1,3,1},50] (* Harvey P. Dale, May 30 2018 *)

PARI

Vec((1 + x + 2*x^2) / ((1 + x^2)*(1 - 2*x^2)) + O(x^60)) \\ Colin Barker, Dec 15 2017

Formula

a(n) = (a(n-1)+1)/2 for n=2, 6, 10...

a(n) = 4*a(n-1)-1 for n=3, 7, 11...

a(n) = (a(n-1)-1)/2 for n=4, 8, 12...

a(n) = 4*a(n-1)+1 for n=5, 9, 13....

From Colin Barker, Dec 15 2017: (Start)

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

a(n) = a(n-2) + 2*a(n-4) for n>3.

(End)

Extensions

Edited by Don Reble, Jan 25 2006