A118391 Numerator of sum of reciprocals of first n tetrahedral numbers A000292.

1, 5, 27, 7, 10, 81, 35, 22, 81, 65, 77, 135, 52, 119, 405, 76, 85, 567, 209, 115, 378, 275, 299, 486, 175, 377, 1215, 217, 232, 1485, 527, 280, 891, 629, 665, 1053, 370, 779, 2457, 430, 451, 2835, 989, 517, 1620, 1127, 1175, 1836, 637, 1325, 4131, 715, 742

Offset

1,2

Comments

Denominators are A118392. Fractions are: 1/1, 5/4, 27/20, 7/5, 10/7, 81/56, 35/24, 22/15, 81/55, 65/44, 77/52, 135/91, 52/35, 119/80, 405/272, 76/51, 85/57, 567/380, 209/140, 115/77, 378/253, 275/184, 299/200, 486/325, 175/117, 377/252, 1215/812, 217/145, 232/155, 1485/992.

2n+3 divides a(2n). 2n-1 divides a(2n-1). p divides a(p) for prime p>2. The only primes in a(n) are a(2) = 5 and a(4) = 7. - Alexander Adamchuk, May 08 2007

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

A118391(n)/A118392(n) = Sum_{i=1..n} 1/A000292(n).

A118391(n)/A118392(n) = Sum_{i=1..n} 1/C(n+2,3).

A118391(n)/A118392(n) = Sum_{i=1..n} 6/(n*(n+1)*(n+2)).

a(n) = Numerator( 3*n*(n+3)/(2*(n+1)*(n+2)) ). - Alexander Adamchuk, May 08 2007

Example

a(1) = 1 = numerator of 1/1.
a(2) = 5 = numerator of 5/4 = 1/1 + 1/4.
a(3) = 27 = numerator of 27/20 = 1/1 + 1/4 + 1/10.
a(4) = 7 = numerator of 7/5 = 1/1 + 1/4 + 1/10 + 1/20.
a(5) = 10 = numerator of 10/7 = 1/1 + 1/4 + 1/10 + 1/20 + 1/35.
a(20) = 115 = numerator of 115/77 = 1/1 + 1/4 + 1/10 + 1/20 + 1/35 + 1/56 + 1/84 + 1/120 + 1/165 + 1/220 + 1/286 + 1/364 + 1/455 + 1/560 + 1/680 + 1/816 + 1/969 + 1/1140 + 1/1330 + 1/1540.

Maple

A118391:= n-> numer(3*n*(n+3)/(2*(n+1)*(n+2))); seq(A118391(n), n=1..60) # G. C. Greubel, Feb 18 2021

Mathematica

Table[ Numerator[3n(n+3)/(2(n+1)(n+2))], {n, 1, 100} ] (* Alexander Adamchuk, May 08 2007 *)
Accumulate[1/Binomial[Range[60]+2, 3]]//Numerator (* Harvey P. Dale, Aug 31 2023 *)

Prog

(PARI) s=0; for(i=3, 50, s+=1/binomial(i, 3); print(numerator(s))) /* Phil Carmody, Mar 27 2012 */
(Sage) [numerator(3*n*(n+3)/(2*(n+1)*(n+2))) for n in (1..60)] # G. C. Greubel, Feb 18 2021
(Magma) [Numerator(3*n*(n+3)/(2*(n+1)*(n+2))): n in [1..60]]; // G. C. Greubel, Feb 18 2021

Crossrefs

Cf. A000292, A022998, A118392.

Sequence in context: A366332 A342998 A342997 * A196085 A097088 A196009

Adjacent sequences: A118388 A118389 A118390 * A118392 A118393 A118394

Keyword

easy,frac,nonn

Author

Jonathan Vos Post, Apr 27 2006

Extensions

More terms from Alexander Adamchuk, May 08 2007

Status

approved