A189642 Numerator of H(n+4) - H(n), where H(n) = Sum_{k=1..n} 1/k.

25, 77, 19, 319, 533, 275, 1207, 1691, 763, 3013, 3875, 543, 6061, 7409, 2981, 10675, 12617, 4927, 17179, 19823, 2525, 25897, 29351, 11033, 37153, 41525, 15409, 51271, 56669, 6937, 68575, 75107, 27347, 89389, 97163, 35125, 114037, 123161, 14751, 142843, 153425

Offset: 0

Author

Gary Detlefs, May 02 2011

Keywords

• nonn
• easy

Comments

a(n) = Numerator of (4*n^3+30*n^2+70*n+50)/((n+1)*(n+2)*(n+3)*(n+4)).

(4*n^3+30*n^2+70*n+50)/a(n) has period length 9, repeating 1, 1, 9, 1, 1, 3, 1, 1, 3.

It is of interest to note that the roots of 4*n^3+30*n^2+70*n+50 are -phi-2, phi-3, and -5/2, where phi = (1+sqrt(5))/2.

H(n+4)-H(n) = (2*n^3+15*n^2+35*n+25)/(12*binomial(n+4,4)).

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A001008, A002805.

Programs

Magma

Harmonic:=func< n | n eq 0 select 0 else &+[ 1/k: k in [1..n] ] >; A189642:=func< n | Numerator( Harmonic(n+4)-Harmonic(n) ) >; [ Numerator( A189642(n) ): n in [0..40] ]; // Klaus Brockhaus, May 21 2011

Maple

h:=n-> sum(1/k,k=1..n):seq(numer(h(n+4)-h(n)), n=0..30);
# alternative Maple program:
P:=(k,n)-> floor(1/2*cos(2*n*Pi/k)+3/5):
seq((4*n^3+30*n^2+70*n+50)/(2*(1+2*P(3,n+1))*(1+2*P(9,n+7))),n=0..30);

Mathematica

Numerator[Table[HarmonicNumber[n+4] - HarmonicNumber[n], {n, 0, 100}]] (* T. D. Noe, May 24 2011 *)

Formula

a(n) = (4*n^3+30*n^2+70*n+50)/(2*(1+2*P(3, n+1))*(1+2*P(9, n+7))), where P(k, n) = floor(1/2*cos(2*n*Pi/k)+3/5).