OFFSET
0,2
COMMENTS
This is an auxiliary sequence for computing A138606.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).
FORMULA
Without reference to A000045: a(n)=2*Floor(a(n-1)/2)+a(n-2). - Clark Kimberling, Nov 07 2009
If n mod 2 = 0 then a(n) = a(n-1) + a(n-2), else a(n) = a(n-1) + a(n-2) - 1.
a(n) = 2*Fibonacci(n) + (1+(-1)^n)/2.
a(n) = 2*Fibonacci(n) + [(n+1)mod 2]. - Gary Detlefs, Dec 29 2010
G.f.: (1 + x - x^2 - 2*x^3)/((1 - x^2)*(1 - x - x^2)). - Ilya Gutkovskiy, Apr 22 2016
From Colin Barker, Apr 22 2016: (Start)
a(n) = a(n-1)+2*a(n-2)-a(n-3)-a(n-4) for n>3.
a(n) = (1/2+(-1)^n/2-(2*((1/2*(1-sqrt(5)))^n-(1/2*(1+sqrt(5)))^n))/sqrt(5)).
(End)
MATHEMATICA
Table[2*Fibonacci[n] + (1 + (-1)^n)/2, {n, 0, 100}] (* G. C. Greubel, Apr 21 2016 *)
LinearRecurrence[{1, 2, -1, -1}, {1, 2, 3, 4}, 40] (* Harvey P. Dale, May 01 2018 *)
PROG
(PARI) Vec((1+x-x^2-2*x^3)/((1-x)*(1+x)*(1-x-x^2)) + O(x^50)) \\ Colin Barker, Apr 22 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Antti Karttunen, Oct 05 2009
STATUS
approved