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A112387
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a(n) = 2^(n/2) if n is even and a(n-1) - a(n-2) if n is odd, a(1) = 1.
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4
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
G.f.: (1+x+x^2)/((1+x^2)*(1-2*x^2)). - Joerg Arndt, Apr 25 2021
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MAPLE
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a:= proc(n) option remember;
`if`(n::even, 2^(n/2), a(n-1)-a(n-2))
end: a(1):=1:
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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