A083416 Add 1, double, add 1, double, etc.

1, 2, 4, 5, 10, 11, 22, 23, 46, 47, 94, 95, 190, 191, 382, 383, 766, 767, 1534, 1535, 3070, 3071, 6142, 6143, 12286, 12287, 24574, 24575, 49150, 49151, 98302, 98303, 196606, 196607, 393214, 393215, 786430, 786431, 1572862, 1572863, 3145726, 3145727, 6291454

Offset: 1

Author

N. J. A. Sloane, Jun 10 2003

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-2).

Crossrefs

Cf. A033484, A081026, A153893.

Programs

Haskell

a083416 n = a083416_list !! (n-1)
a083416_list = 1 : f 2 1 where
   f x y = z : f (x+1) z where z = (1 + x `mod` 2) * y + 1 - x `mod` 2
-- Reinhard Zumkeller, Feb 27 2012

Magma

[Floor(3*2^((2*n-(-1)^n-3)/4)+((-1)^n-3)/2): n in [1..50]]; // Vincenzo Librandi, Aug 17 2011

Maple

A083416 := proc(n) if type(n,'even') then 3*2^(n/2-1)-1 ; else 3*2^((n-1)/2)-2 ; end if; end proc: # R. J. Mathar, Feb 16 2011

Mathematica

a=0; b=0; lst={a,b}; Do[z=a+b+1; AppendTo[lst,z]; a=b; b=z; z=b+1; AppendTo[lst,z]; a=b; b=z,{n,50}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 16 2010 *)
LinearRecurrence[{0,3,0,-2},{1,2,4,5},40] (* Harvey P. Dale, Nov 18 2014 *)

Formula

G.f.: x*(1+2*x+x^2-x^3)/(1-x^2)/(1-2*x^2).

a(2*n) = 3*2^(n-1)-1, a(2*n+1) = 3*2^n-2.

a(n) = A081026(n+1)-1.

a(n) = 3*2^((2*n-(-1)^n-3)/4)+((-1)^n-3)/2. - Bruno Berselli, Feb 17 2011

For n > 1: a(n) = (1 + n mod 2) * a(n-1) + 1 - n mod 2. - Reinhard Zumkeller, Feb 27 2012

a(2n+1) = A033484(n), a(2n) = A153893(n). - Philippe Deléham, Apr 14 2013

E.g.f.: (3*cosh(sqrt(2)*x) - 4*sinh(x) + 3*sqrt(2)*sinh(sqrt(2)*x) - 2*cosh(x) - 1)/2. - Stefano Spezia, Jul 11 2023

Extensions

More terms from Donald Sampson (marsquo(AT)hotmail.com), Dec 04 2003

Corrected by T. D. Noe, Nov 02 2006