A112387 a(n) = 2^(n/2) if n is even and a(n-1) - a(n-2) if n is odd, a(1) = 1.

1, 1, 2, 1, 4, 3, 8, 5, 16, 11, 32, 21, 64, 43, 128, 85, 256, 171, 512, 341, 1024, 683, 2048, 1365, 4096, 2731, 8192, 5461, 16384, 10923, 32768, 21845, 65536, 43691, 131072, 87381, 262144, 174763, 524288, 349525, 1048576, 699051, 2097152, 1398101, 4194304

Offset: 0

Author

Edwin F. Sampang, Dec 05 2005

Keywords

• nonn
• easy

Comments

This sequence originated from the Fibonacci sequence, but instead of adding the last two terms, you get the average. Example, if you have the initial condition a(1)=x and a(2)=y, a(3)=(x+y)/2, a(4)=(x+3y)/4, a(5)=(3x+5y)/8, a(6)=(5x+11y)/16 and so on and so forth. I used the coefficients of x and y as well as the denominator.

As n approaches infinity a(n)/a(n+1) oscillates between the values 3/2 and 1/3.

Links

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

Crossrefs

Cf. A000079, A001045, A048573, A135318.

Programs

Maple

a:= proc(n) option remember;
      `if`(n::even, 2^(n/2), a(n-1)-a(n-2))
    end: a(1):=1:
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 27 2023

Mathematica

a[1] = 1; a[2] = 2; a[n_] := a[n] = If[ EvenQ[n], 2^(n/2), a[n - 1] - a[n - 2]]; Array[a, 43] (* Robert G. Wilson v, Dec 05 2005 *)
nxt[{n_,a_,b_}]:={n+1,b,If[OddQ[n],2^((n+1)/2),b-a]}; NestList[nxt,{2,1,2},50][[All,2]] (* Harvey P. Dale, Jul 08 2019 *)

Formula

a(n) = 2^(n/2) if n is even, a(n) = a(n-1) - a(n-2) if n is odd, and a(1) = 1.

a(2n) = A000079(n), a(2n-1) = A001045(n).

G.f.: (1+x+x^2)/((1+x^2)*(1-2*x^2)). - Joerg Arndt, Apr 25 2021

a(n) = A135318(n + (-1)^n). - Paul Curtz, Sep 27 2023

E.g.f.: (3*cosh(sqrt(2)*x) + sin(x) + sqrt(2)*sinh(sqrt(2)*x))/3. - Stefano Spezia, Jun 30 2024

a(2*n) + a(2*n+1) = A048573(n); a(2*n+1) + a(2*n+2) = A001045(n+3). - Paul Curtz, Jan 03 2025

Extensions

Edited and extended by Robert G. Wilson v, Dec 05 2005

a(0)=1 prepended by Alois P. Heinz, Sep 27 2023