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A188386
a(n) = numerator(H(n+2)-H(n-1)), where H(n) = Sum_{k=1..n} 1/k is the n-th harmonic number.
7
11, 13, 47, 37, 107, 73, 191, 121, 299, 181, 431, 253, 587, 337, 767, 433, 971, 541, 1199, 661, 1451, 793, 1727, 937, 2027, 1093, 2351, 1261, 2699, 1441, 3071, 1633, 3467, 1837, 3887, 2053, 4331, 2281, 4799, 2521, 5291, 2773, 5807, 3037, 6347, 3313, 6911, 3601
OFFSET
1,1
COMMENTS
Denominators are listed in A033931.
A027446 appears to be divisible by a(n).
The sequence lists also the largest odd divisors of 3*m^2-1 (A080663) for m>1. In fact, for m even, the largest odd divisor is 3*m^2-1 itself; for m odd, the largest odd divisor is (3*m^2-1)/2. From this follows the second formula given in Formula field. - Bruno Berselli, Aug 27 2013
FORMULA
a(n) = numerator((3*n^2+6*n+2)/(n*(n+1)*(n+2))).
a(n) = (3-(-1)^n)*(3*n^2+6*n+2)/4.
a(2n+1) = A158463(n+1), a(2n) = A003154(n+1).
G.f.: -x*(11+13*x+14*x^2-2*x^3-x^4+x^5) / ( (x-1)^3*(1+x)^3 ). - R. J. Mathar, Apr 09 2011
a(n) = numerator of coefficient of x^3 in the Maclaurin expansion of sin(x)*exp((n+1)*x). - Francesco Daddi, Aug 04 2011
H(n+3) = 3/2 + 2*f(n)/((n+2)*(n+3)), where f(n) = Sum_{k=0..n}((-1)^k*binomial(-3,k)/(n+1-k)). - Gary Detlefs, Jul 17 2011
a(n) = A213998(n+2,2). - Reinhard Zumkeller, Jul 03 2012
Sum_{n>=1} 1/a(n) = c*(tan(c) - cot(c)/2) - 1/2, where c = Pi/(2*sqrt(3)). - Amiram Eldar, Sep 27 2022
MAPLE
seq((3-(-1)^n)*(3*n^2+6*n+2)/4, n=1..100);
MATHEMATICA
Table[(3 - (-1)^n)*(3*n^2 + 6*n + 2)/4, {n, 40}] (* Wesley Ivan Hurt, Jan 29 2017 *)
Numerator[#[[4]]-#[[1]]]&/@Partition[HarmonicNumber[Range[0, 50]], 4, 1] (* or *) LinearRecurrence[{0, 3, 0, -3, 0, 1}, {11, 13, 47, 37, 107, 73}, 50] (* Harvey P. Dale, Dec 31 2017 *)
PROG
(Magma) [Numerator((3*n^2+6*n+2)/((n*(n+1)*(n+2)))): n in [1..50]]; // Vincenzo Librandi, Mar 30 2011
(Haskell)
import Data.Ratio ((%), numerator)
a188386 n = a188386_list !! (n-1)
a188386_list = map numerator $ zipWith (-) (drop 3 hs) hs
where hs = 0 : scanl1 (+) (map (1 %) [1..])
-- Reinhard Zumkeller, Jul 03 2012
CROSSREFS
KEYWORD
nonn,easy,look
AUTHOR
Gary Detlefs, Mar 29 2011
STATUS
approved