cp's OEIS Frontend

A045896 Denominator of n/((n+1)*(n+2)) = A026741/A045896.

%I A045896 #66 Apr 24 2024 19:52:15
%S A045896 1,6,6,20,15,42,28,72,45,110,66,156,91,210,120,272,153,342,190,420,
%T A045896 231,506,276,600,325,702,378,812,435,930,496,1056,561,1190,630,1332,
%U A045896 703,1482,780,1640,861,1806,946,1980,1035,2162,1128,2352,1225,2550,1326,2756,1431
%N A045896 Denominator of n/((n+1)*(n+2)) = A026741/A045896.
%C A045896 Also period length divided by 2 of pairs (a,b), where a has period 2*n-2 and b has period n.
%C A045896 From _Paul Curtz_, Apr 17 2014: (Start)
%C A045896 Difference table of A026741/A045896:
%C A045896    0,      1/6,    1/6,    3/20,     2/15,    5/42, ...
%C A045896    1/6,      0,  -1/60,   -1/60,    -1/70,   -1/84, ... = 1/6, -A051712/A051713
%C A045896   -1/6,  -1/60,      0,   1/420,    1/420,   1/504, ...
%C A045896    3/20,  1/60,  1/420,       0,  -1/2520, -1/2520, ...
%C A045896   -2/15, -1/70, -1/420, -1/2520,        0, 1/13860, ...
%C A045896    5/42,  1/84,  1/504,  1/2520, -1/13860,       0, ...
%C A045896 Autosequence of the first kind. The main diagonal is A000004. The first two upper diagonals are equal. Their denominators are A000911. (End)
%H A045896 Reinhard Zumkeller, <a href="/A045896/b045896.txt">Table of n, a(n) for n = 0..10000</a>
%H A045896 Ralf W. Grosse-Kunstleve, <a href="https://web.archive.org/web/20070603212504/http://cci.lbl.gov/~rwgk/EIS/Ceiling.html">Origin of EIS sequences A045895 & A045896</a>. [Wayback Machine copy]
%H A045896 Masanobu Kaneko, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/KANEKO/AT-kaneko.html">The Akiyama-Tanigawa algorithm for Bernoulli numbers</a>, J. Integer Sequences, 3 (2000), Article 00.2.9.
%F A045896 G.f.: (2*x^3+3*x^2+6*x+1)/(1-x^2)^3.
%F A045896 a(n) = (n+1)*(n+2) if n odd; or (n+1)*(n+2)/2 if n even = (n+1)*(n+2)*(3-(-1)^n)/4. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 16 2004
%F A045896 a(2*n) = A000384(n+1); a(2*n+1) = A026741(n+1). - _Reinhard Zumkeller_, Dec 12 2011
%F A045896 Sum_{n>=0} 1/a(n) = 1 + log(2). - _Amiram Eldar_, Sep 11 2022
%F A045896 From _Amiram Eldar_, Sep 14 2022: (Start)
%F A045896 a(n) = lcm(2*n+2, n+2)/2.
%F A045896 a(n) = A045895(n+2)/2. (End)
%F A045896 E.g.f.: (2 + 8*x + x^2)*cosh(x)/2 + (2 + 2*x + x^2)*sinh(x). - _Stefano Spezia_, Apr 24 2024
%p A045896 seq((n+1)*(n+2)*(3-(-1)^n)/4, n=0..20); # C. Ronaldo
%p A045896 with(combinat): seq(lcm(n+1,binomial(n+2,n)), n=0..50); # _Zerinvary Lajos_, Apr 20 2008
%t A045896 Table[LCM[2*n + 2, n + 2]/2, {n, 0, 40}] (* corrected by _Amiram Eldar_, Sep 14 2022 *)
%t A045896 Denominator[#[[1]]/(#[[2]]#[[3]])&/@Partition[Range[0,60],3,1]] (* _Harvey P. Dale_, Aug 15 2013 *)
%o A045896 (Haskell)
%o A045896 import Data.Ratio ((%), denominator)
%o A045896 a045896 n = denominator $ n % ((n + 1) * (n + 2))
%o A045896 -- _Reinhard Zumkeller_, Dec 12 2011
%o A045896 (PARI) Vec((2*x^3+3*x^2+6*x+1)/(1-x^2)^3+O(x^99)) \\ _Charles R Greathouse IV_, Mar 23 2016
%Y A045896 Factor of A160466.
%Y A045896 Cf. A000004, A000384, A000911, A045895, A026741, A051714, A241269.
%K A045896 nonn,easy,frac,nice
%O A045896 0,2
%A A045896 _Ralf W. Grosse-Kunstleve_, Dec 11 1999