A271535 a(n) = ( n*(n + 1)*(2*n + 1)/6 )^2.
0, 1, 25, 196, 900, 3025, 8281, 19600, 41616, 81225, 148225, 256036, 422500, 670761, 1030225, 1537600, 2238016, 3186225, 4447881, 6100900, 8236900, 10962721, 14402025, 18696976, 24010000, 30525625, 38452401, 48024900, 59505796, 73188025, 89397025, 108493056
Offset: 0
Links
- R. D. Carmichael and T. L. DeLand, Find the sum of the series 1^2 + 5^2 + 14^2 + 30^2 + ... + [n*(n+1)*(2*n+1)/6]^2, American Mathematical Monthly, Vol. 15, No. 6/7, Jun-Jul, 1908, pp. 132-133.
- Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
Programs
-
Magma
[(n*(n+1)*(2*n+1)/6)^2: n in [0..50]];
-
Mathematica
Table[(n (n + 1) (2 n + 1)/6)^2, {n, 0, 50}] -
PARI
vector(100, n, n--; (n*(n + 1)*(2*n + 1)/6)^2) \\ Altug Alkan, Apr 21 2016
Formula
G.f.: x*(1 + 18*x + 42*x^2 + 18*x^3 + x^4)/(1 - x)^7.
a(n) = Sum_{j=1..n} Sum_{i=1..n} (i*j)^2. - Alexander Adamchuk, Oct 26 2004
E.g.f.: x*(36 + 414*x + 744*x^2 + 393*x^3 + 72*x^4 + 4*x^5)*exp(x)/36. - Ilya Gutkovskiy, Apr 21 2016
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
Sum_{i = 0..n} a(i) = n*(n + 1)*(n + 2)*(2*n + 1)*(2*n + 3)*(5*n^2 + 10*n - 1)/1260. [See Carmichael - DeLand in Links section, page 132.]
a(n) = A000330(n)^2. - Ray Chandler, Apr 21 2016
Sum_{n>=1} 1/a(n) = 84*Pi^2 - 828. - Amiram Eldar, Feb 25 2023
Extensions
Edited by Bruno Berselli, Apr 22 2016