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A112387
a(n) = 2^(n/2) if n is even and a(n-1) - a(n-2) if n is odd, a(1) = 1.
7
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.
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
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