This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A273692 #32 Jan 05 2025 19:51:40 %S A273692 2,8,2,32,32,128,64,512,512,2048,128,8192,8192,32768,16384,131072, %T A273692 131072,524288,131072,2097152,2097152,8388608,4194304,33554432, %U A273692 33554432,134217728,16777216,536870912,536870912,2147483648,1073741824,8589934592,8589934592,34359738368 %N A273692 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 A273692 O(n) is the Horadam notation. %C A273692 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 A273692 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 A273692 Differences table of Oco(n): %C A273692 1/2, 5/8, 1/2, 11/32, 7/32, 17/128, 5/64, ... %C A273692 1/8, -1/8, -5/32, -1/8, -11/128, -7/128, ... %C A273692 -1/4, -1/32, 1/32, 5/128, 1/32, ... %C A273692 7/32, 1/16, 1/128, -1/128, ... %C A273692 -5/32, -7/128, -1/64, ... %C A273692 13/128, 5/128, ... %C A273692 -1/16, ... . %C A273692 First column: Io(n) = 1/2 followed by (-1)^n* A067745(n)/(8, 4, 32, 32, ...). %C A273692 1) Alternated Oco(2n) + Io(2n) and Oco(2n+1) - Io(2n+1) gives 2^n. %C A273692 2) Alternated Oco(2n) - Io(2n) and Oco(2n+1) + Io(2n+1) gives 3*O(n)/2. %C A273692 (1/2 - 1/2 = 0, 5/8 + 1/8 = 3/4, 1/2 + 1/4 = 3/4, 11/32 + 7/32 = 9/16, ...) %D A273692 M. R. Bacon and C. K. Cook, Some properties of Oresme numbers and convolutions ..., Fib. Q., 62:3 (2024), 233-240. %H A273692 Seiichi Manyama, <a href="/A273692/b273692.txt">Table of n, a(n) for n = 0..3320</a> %H A273692 A. F. Horadam, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/12-3/horadam.pdf">Oresme numbers</a>, Fib. Quart., 12 (1974), 267-271. %F A273692 a(n) = denominator of (2+3*n)/2^(n+2). %F A273692 a(2n+1) = 8*4^n. %F A273692 a(2n+2) = a(2n+1)/(4, 1, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, ..., shifted A006519?). %o A273692 (PARI) Or(n) = n/2^n; %o A273692 a(n) = denominator(2*Or(n+1) - Or(n+2)); \\ _Michel Marcus_, May 28 2016 %Y A273692 Cf. A000079, A000265, A000302, A004171 (main diagonal), A006519, A016789, A067745, A075101, A075677, A209308. %K A273692 nonn,frac %O A273692 0,1 %A A273692 _Paul Curtz_, May 28 2016 %E A273692 More terms from _Michel Marcus_, May 28 2016