OFFSET
0,3
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).
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)
MATHEMATICA
LinearRecurrence[{0, 1, 0, 2}, {1, 1, 3, 1}, 50] (* Harvey P. Dale, May 30 2018 *)
PROG
(PARI) Vec((1 + x + 2*x^2) / ((1 + x^2)*(1 - 2*x^2)) + O(x^60)) \\ Colin Barker, Dec 15 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Edwin F. Sampang, Dec 12 2005
EXTENSIONS
Edited by Don Reble, Jan 25 2006
STATUS
approved