A099903 Sum of all matrix elements of n X n matrix M(i,j) = i^3+j^3, (i,j = 1..n). a(n) = n^3*(n+1)^2/2.
2, 36, 216, 800, 2250, 5292, 10976, 20736, 36450, 60500, 95832, 146016, 215306, 308700, 432000, 591872, 795906, 1052676, 1371800, 1764000, 2241162, 2816396, 3504096, 4320000, 5281250, 6406452, 7715736, 9230816, 10975050, 12973500
Offset: 1
Examples
a(3) = (1/2) * (2^3)*(2+1)^2 = 36. (or) a(3) = (1^3+1^3) + (1^3+2^3) + (2^3+1^3) + (2^3+2^3) = 36.
Links
- Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).
Programs
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Magma
[(n^5+2*n^4+n^3)/2: n in [1..30]]; // Wesley Ivan Hurt, May 25 2014
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Maple
A099903:=n->(n^5+2*n^4+n^3)/2; seq(A099903(n), n=1..30); # Wesley Ivan Hurt, Feb 26 2014
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Mathematica
Table[(n^5+2*n^4+n^3)/2, {n, 30}] (* Wesley Ivan Hurt, Feb 26 2014 *) -
PARI
a(n)=(n^5+2*n^4+n^3)/2 \\ Charles R Greathouse IV, Oct 27 2011
Formula
a(n) = Sum_{i=1..n, j=1..n} (i^3 + j^3).
a(n) = 2*n*Sum_{k=1..n} k^3. - Gary Detlefs, Oct 26 2011
a(n) = (n^5 + 2*n^4 + n^3)/2. - Charles R Greathouse IV, Oct 27 2011
G.f.: 2*x*(1+12*x+15*x^2+2*x^3)/(1-x)^6. - Colin Barker, May 04 2012
From Amiram Eldar, Nov 02 2021: (Start)
Sum_{n>=1} 1/a(n) = 2*zeta(3) - Pi^2 + 8.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*zeta(3)/2 + 12*log(2) - Pi^2/6 - 8. (End)
Comments