A166012 a(n) = 2*(A000045(n)-(n mod 2)) + 1 + (n mod 2).

1, 2, 3, 4, 7, 10, 17, 26, 43, 68, 111, 178, 289, 466, 755, 1220, 1975, 3194, 5169, 8362, 13531, 21892, 35423, 57314, 92737, 150050, 242787, 392836, 635623, 1028458, 1664081, 2692538, 4356619, 7049156, 11405775, 18454930, 29860705, 48315634, 78176339

Offset: 0

Author

Antti Karttunen, Oct 05 2009

Keywords

• nonn
• easy

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).

Programs

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 *)

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

Formula

a(2n) = 2*A000045(2n) + 1, a(2n+1) = 2*A000045(2n+1).

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)