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A273692
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a(n) is the denominator of 2*O(n+1) - O(n+2) where O(n) = n/2^n, the n-th Oresme number.
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1
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2, 8, 2, 32, 32, 128, 64, 512, 512, 2048, 128, 8192, 8192, 32768, 16384, 131072, 131072, 524288, 131072, 2097152, 2097152, 8388608, 4194304, 33554432, 33554432, 134217728, 16777216, 536870912, 536870912, 2147483648, 1073741824, 8589934592, 8589934592, 34359738368
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OFFSET
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0,1
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COMMENTS
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O(n) is the Horadam notation.
O(n) or Oresme(n) = n/2^n = 0, 1/2, 1/2, 3/8, 1/4, ... . The positive Oresme numbers are O(n+1) = A000265(n+1)/A075101(n+1). See A209308. Consider Oco(n) = 2*O(n+1) - O(n+2) = 1/2, 5/8, 1/2, 11/32, 7/32, ... = A075677(n+1)/a(n). (See Coll(n) in A209308.)
Oco(n) = 1/2, 5/8, 1/2, 11/32, 7/32, 17/128, 5/64, 23/512, 13/512, 29/2048, 1/128, 35/8192, 19/8192, ... . Compare to (2+3*n)/2^(n+2).
Differences table of Oco(n):
1/2, 5/8, 1/2, 11/32, 7/32, 17/128, 5/64, ...
1/8, -1/8, -5/32, -1/8, -11/128, -7/128, ...
-1/4, -1/32, 1/32, 5/128, 1/32, ...
7/32, 1/16, 1/128, -1/128, ...
-5/32, -7/128, -1/64, ...
13/128, 5/128, ...
-1/16, ... .
First column: Io(n) = 1/2 followed by (-1)^n* A067745(n)/(8, 4, 32, 32, ...).
1) Alternated Oco(2n) + Io(2n) and Oco(2n+1) - Io(2n+1) gives 2^n.
2) Alternated Oco(2n) - Io(2n) and Oco(2n+1) + Io(2n+1) gives 3*O(n)/2.
(1/2 - 1/2 = 0, 5/8 + 1/8 = 3/4, 1/2 + 1/4 = 3/4, 11/32 + 7/32 = 9/16, ...)
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LINKS
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FORMULA
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a(n) = denominator of (2+3*n)/2^(n+2).
a(2n+1) = 8*4^n.
a(2n+2) = a(2n+1)/(4, 1, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, ..., shifted A006519?).
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PROG
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(PARI) Or(n) = n/2^n;
a(n) = denominator(2*Or(n+1) - Or(n+2)); \\ Michel Marcus, May 28 2016
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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