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.

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

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

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

Cf. A033931 (denominators), A001008, A001711, A027446, A002805, A003154, A080663, A158463, A213998.

Sequence in context: A108264 A023268 A154292 * A027450 A234799 A036295

Adjacent sequences: A188383 A188384 A188385 * A188387 A188388 A188389

Keyword

nonn,easy,look

Author

Gary Detlefs, Mar 29 2011

Status

approved