A098077 a(n) = n^2*(n+1)*(2*n+1)/3.

2, 20, 84, 240, 550, 1092, 1960, 3264, 5130, 7700, 11132, 15600, 21294, 28420, 37200, 47872, 60690, 75924, 93860, 114800, 139062, 166980, 198904, 235200, 276250, 322452, 374220, 431984, 496190, 567300, 645792, 732160, 826914, 930580, 1043700

Offset

1,1

Comments

Sum of all matrix elements M(i,j) = i^2 + j^2 (i,j = 1,...,n).

From Torlach Rush, Jan 05 2020: (Start)

a(n) = n * A006331(n).

tr(M(n)) = A006331(n).

The sum of the antidiagonal of M(n) equals tr(M(n)).

M(n) = M(n)' (Symmetric).

M(1,) = M(,1) = A002522(n), n > 0.

M(2,) = M(,2) = A087475(n), n > 0.

M(3,) = M(,3) = A189834(n), n > 0.

M(4,) = M(,4) = A241751(n), n > 0.

(End)

Consider the partitions of 2n into two parts (p,q) where p <= q. Then a(n) is the total volume of the family of rectangular prisms with dimensions p, p and p+q. - Wesley Ivan Hurt, Apr 15 2018

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = Sum_{j=1..n} Sum_{i=1..n} (i^2 + j^2).

G.f.: 2*x*(1 + 5*x + 2*x^2)/(1-x)^5. - Colin Barker, May 04 2012

E.g.f.: (1/3)*exp(x)*x*(6 + 24*x + 15*x^2 + 2*x^3) . - Stefano Spezia, Jan 06 2020

a(n) = a(n-1) + (8*n^3 - 3*n^2 + n)/3. - Torlach Rush, Jan 07 2020

From Amiram Eldar, May 31 2022: (Start)

Sum_{n>=1} 1/a(n) = Pi^2/2 + 24*log(2) - 21.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/4 - 6*Pi - 6*log(2) + 21. (End)

From G. C. Greubel, Apr 09 2023: (Start)

a(n) = (1/4)*A100431(n-1).

a(n) = 2*A108678(n-1). (End)

Example

a(2) = (1^2 + 1^2) + (1^2 + 2^2) + (2^2 + 1^2) + (2^2 + 2^2) = 2 + 5 + 5 + 8 = 20.

Mathematica

Table[ Sum[i^2 + j^2, {i, n}, {j, n}], {n, 35}]
LinearRecurrence[{5, -10, 10, -5, 1}, {2, 20, 84, 240, 550}, 40] (* Vincenzo Librandi, Apr 16 2018 *)

Prog

(PARI) a(n)=n^2*(n+1)*(2*n+1)/3 \\ Charles R Greathouse IV, Oct 07 2015
(Magma) [n^2*(n+1)*(2*n+1)/3: n in [1..40]]; // G. C. Greubel, Apr 09 2023
(SageMath) [n^2*(n+1)*(2*n+1)/3 for n in range(1, 41)] # G. C. Greubel, Apr 09 2023

Crossrefs

Cf. A002522, A006331, A087475, A100431, A108678, A189834, A241751.

Sequence in context: A135188 A161007 A331760 * A279264 A063663 A071253

Adjacent sequences: A098074 A098075 A098076 * A098078 A098079 A098080

Keyword

nonn,easy

Author

Alexander Adamchuk, Oct 24 2004

Extensions

More terms from Robert G. Wilson v, Nov 01 2004

New definition from Ralf Stephan, Dec 01 2004

Status

approved