%I #30 Sep 20 2024 09:57:03
%S 2,8,2,32,32,128,64,512,512,2048,128,8192,8192,32768,16384,131072,
%T 131072,524288,131072,2097152,2097152,8388608,4194304,33554432,
%U 33554432,134217728,16777216,536870912,536870912,2147483648,1073741824,8589934592,8589934592,34359738368
%N a(n) is the denominator of 2*O(n+1) - O(n+2) where O(n) = n/2^n, the n-th Oresme number.
%C O(n) is the Horadam notation.
%C 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.)
%C 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).
%C Differences table of Oco(n):
%C 1/2, 5/8, 1/2, 11/32, 7/32, 17/128, 5/64, ...
%C 1/8, -1/8, -5/32, -1/8, -11/128, -7/128, ...
%C -1/4, -1/32, 1/32, 5/128, 1/32, ...
%C 7/32, 1/16, 1/128, -1/128, ...
%C -5/32, -7/128, -1/64, ...
%C 13/128, 5/128, ...
%C -1/16, ... .
%C First column: Io(n) = 1/2 followed by (-1)^n* A067745(n)/(8, 4, 32, 32, ...).
%C 1) Alternated Oco(2n) + Io(2n) and Oco(2n+1) - Io(2n+1) gives 2^n.
%C 2) Alternated Oco(2n) - Io(2n) and Oco(2n+1) + Io(2n+1) gives 3*O(n)/2.
%C (1/2 - 1/2 = 0, 5/8 + 1/8 = 3/4, 1/2 + 1/4 = 3/4, 11/32 + 7/32 = 9/16, ...)
%D M. R. Bacon and C. K. Cook, Some properties of Oresme numbers and convolutions ..., Fib. Q., 62:3 (2024), 233-240.
%H Seiichi Manyama, <a href="/A273692/b273692.txt">Table of n, a(n) for n = 0..3320</a>
%H A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/12-3/horadam.pdf">Oresme numbers</a>, Fib. Quart., 12 (1974), 267-271.
%F a(n) = denominator of (2+3*n)/2^(n+2).
%F a(2n+1) = 8*4^n.
%F a(2n+2) = a(2n+1)/(4, 1, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, ..., shifted A006519?).
%o (PARI) Or(n) = n/2^n;
%o a(n) = denominator(2*Or(n+1) - Or(n+2)); \\ _Michel Marcus_, May 28 2016
%Y Cf. A000079, A000265, A000302, A004171 (main diagonal), A006519, A016789, A067745, A075101, A075677, A209308.
%K nonn,frac
%O 0,1
%A _Paul Curtz_, May 28 2016
%E More terms from _Michel Marcus_, May 28 2016