|
%I #19 Aug 31 2023 17:45:59
%S 1,5,27,7,10,81,35,22,81,65,77,135,52,119,405,76,85,567,209,115,378,
%T 275,299,486,175,377,1215,217,232,1485,527,280,891,629,665,1053,370,
%U 779,2457,430,451,2835,989,517,1620,1127,1175,1836,637,1325,4131,715,742
%N Numerator of sum of reciprocals of first n tetrahedral numbers A000292.
%C 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.
%C 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
%H G. C. Greubel, <a href="/A118391/b118391.txt">Table of n, a(n) for n = 1..1000</a>
%F A118391(n)/A118392(n) = Sum_{i=1..n} 1/A000292(n).
%F A118391(n)/A118392(n) = Sum_{i=1..n} 1/C(n+2,3).
%F A118391(n)/A118392(n) = Sum_{i=1..n} 6/(n*(n+1)*(n+2)).
%F a(n) = Numerator( 3*n*(n+3)/(2*(n+1)*(n+2)) ). - _Alexander Adamchuk_, May 08 2007
%e a(1) = 1 = numerator of 1/1.
%e a(2) = 5 = numerator of 5/4 = 1/1 + 1/4.
%e a(3) = 27 = numerator of 27/20 = 1/1 + 1/4 + 1/10.
%e a(4) = 7 = numerator of 7/5 = 1/1 + 1/4 + 1/10 + 1/20.
%e a(5) = 10 = numerator of 10/7 = 1/1 + 1/4 + 1/10 + 1/20 + 1/35.
%e 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.
%p A118391:= n-> numer(3*n*(n+3)/(2*(n+1)*(n+2))); seq(A118391(n), n=1..60) # _G. C. Greubel_, Feb 18 2021
%t Table[ Numerator[3n(n+3)/(2(n+1)(n+2))], {n,1,100} ] (* _Alexander Adamchuk_, May 08 2007 *)
%t Accumulate[1/Binomial[Range[60]+2,3]]//Numerator (* _Harvey P. Dale_, Aug 31 2023 *)
%o (PARI) s=0;for(i=3,50,s+=1/binomial(i,3);print(numerator(s))) /* _Phil Carmody_, Mar 27 2012 */
%o (Sage) [numerator(3*n*(n+3)/(2*(n+1)*(n+2))) for n in (1..60)] # _G. C. Greubel_, Feb 18 2021
%o (Magma) [Numerator(3*n*(n+3)/(2*(n+1)*(n+2))): n in [1..60]]; // _G. C. Greubel_, Feb 18 2021
%Y Cf. A000292, A022998, A118392.
%K easy,frac,nonn
%O 1,2
%A _Jonathan Vos Post_, Apr 27 2006
%E More terms from _Alexander Adamchuk_, May 08 2007
|
