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

0,3

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).

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

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 *)

Crossrefs

Cf. A000079, A001045, A135318.

Sequence in context: A106625 A275902 A008347 * A370727 A193174 A126084

Adjacent sequences: A112384 A112385 A112386 * A112388 A112389 A112390

Keyword

nonn,easy

Author

Edwin F. Sampang, Dec 05 2005

Extensions

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

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

Status

approved