A188386 - OEIS

%I #62 Sep 27 2022 09:00:44

%S 11,13,47,37,107,73,191,121,299,181,431,253,587,337,767,433,971,541,

%T 1199,661,1451,793,1727,937,2027,1093,2351,1261,2699,1441,3071,1633,

%U 3467,1837,3887,2053,4331,2281,4799,2521,5291,2773,5807,3037,6347,3313,6911,3601

%N a(n) = numerator(H(n+2)-H(n-1)), where H(n) = Sum_{k=1..n} 1/k is the n-th harmonic number.

%C Denominators are listed in A033931.

%C A027446 appears to be divisible by a(n).

%C 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

%H Reinhard Zumkeller, <a href="/A188386/b188386.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).

%F a(n) = numerator((3*n^2+6*n+2)/(n*(n+1)*(n+2))).

%F a(n) = (3-(-1)^n)*(3*n^2+6*n+2)/4.

%F a(2n+1) = A158463(n+1), a(2n) = A003154(n+1).

%F 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

%F a(n) = numerator of coefficient of x^3 in the Maclaurin expansion of sin(x)*exp((n+1)*x). - _Francesco Daddi_, Aug 04 2011

%F 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

%F a(n) = A213998(n+2,2). - _Reinhard Zumkeller_, Jul 03 2012

%F 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

%p seq((3-(-1)^n)*(3*n^2+6*n+2)/4, n=1..100);

%t Table[(3 - (-1)^n)*(3*n^2 + 6*n + 2)/4, {n, 40}] (* _Wesley Ivan Hurt_, Jan 29 2017 *)

%t 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 *)

%o (Magma) [Numerator((3*n^2+6*n+2)/((n*(n+1)*(n+2)))): n in [1..50]]; // _Vincenzo Librandi_, Mar 30 2011

%o (Haskell)

%o import Data.Ratio ((%), numerator)

%o a188386 n = a188386_list !! (n-1)

%o a188386_list = map numerator $ zipWith (-) (drop 3 hs) hs

%o    where hs = 0 : scanl1 (+) (map (1 %) [1..])

%o -- _Reinhard Zumkeller_, Jul 03 2012

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

%K nonn,easy,look

%O 1,1

%A _Gary Detlefs_, Mar 29 2011