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A112387
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a(1)=1, a(2)=2, a(n)= 2^(n/2) if even and a(n-1)-a(n-2) if odd.
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1
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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.
Consider b(n)= 1, a(n) = 1, 1, 2, 1, 4, 3, 8, 5, 16, 11, 32, 21 . 1) A000079 is full, A001045 is without 0. 2) b(2n) and b(2n+1) swapped gives 1, 1, 1, 2, 3, 4, 5, 8 = A135318 . 3) 10*b(2n)+b(2n+1)= 11, 21, 43, 85, 171 = A001045(n+5). 4) b(n) differences = 0, 1, -1, 3, -1, 5, -3, 11, -5, 21 : mixed Jacobsthal -A001045(n), A001045(n+2). See A117576. [From Paul Curtz (bpcrtz(AT)free.fr), Sep 09 2008]
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FORMULA
| a(n)=2^(n/2) if n is even, a(n)=a(n-1)-a(n-2) if n is odd and with initial condition of a(1)=1. The limit of a(n)/a(n+1) as n approaches infinity oscillates at a value of 3/2 and 1/3.
a(2n)=A000079(n), a(2n-1)=A001045(n).
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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 *)
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CROSSREFS
| Sequence in context: A128280 A106625 A008347 * A193174 A076077 A152194
Adjacent sequences: A112384 A112385 A112386 * A112388 A112389 A112390
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KEYWORD
| nonn
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AUTHOR
| Edwin F. Sampang (efs_files(AT)yahoo.com), Dec 05 2005
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EXTENSIONS
| Edited and extended by Robert G. Wilson v (rgwv(at)rgwv.com), Dec 05 2005
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