A072474 Sum of next n squares.

1, 13, 77, 294, 855, 2071, 4403, 8492, 15189, 25585, 41041, 63218, 94107, 136059, 191815, 264536, 357833, 475797, 623029, 804670, 1026431, 1294623, 1616187, 1998724, 2450525, 2980601, 3598713, 4315402, 5142019, 6090755, 7174671, 8407728, 9804817, 11381789, 13155485

Offset

1,2

Links

Kelvin Voskuijl, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

a(n) = k*(k+1)*(2*k+1)/6 - r*(r+1)*(2*r+1)/6, where k = n*(n+1)/2 and r = n*(n-1)/2.

a(n) = A000330(n*(n+1)/2) - A000330(n*(n-1)/2).

a(n) = (n/12)*(3*n^2 + 1)*(n^2 + 2). - Benoit Cloitre, Jun 26 2002

G.f.: x*(1+3*x+x^2)*(1+4*x+x^2)/(1-x)^6. - Colin Barker, Mar 23 2012

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n > 6. - Jinyuan Wang, May 25 2020

E.g.f.: exp(x)*x*(12 + 66*x + 82*x^2 + 30*x^3 + 3*x^4)/12. - Stefano Spezia, May 14 2024

Example

a(1) = 1^2 = 1;
a(2) = 2^2 + 3^2 = 13;
a(3) = 4^2 + 5^2 + 6^2 = 77.

Mathematica

Table[Sum[ i^2, {i, n(n - 1)/2 + 1, n(n + 1)/2}], {n, 1, 35}]

Prog

(PARI) a(n) = n*(3*n^2+1)*(n^2+2)/12
(Magma) [n*(3*n^2+1)*(n^2+2)/12: n in [1..35]]; // Vincenzo Librandi, Dec 31 2024

Crossrefs

Cf. A000290, A000330, A072475, A050410.

Cf. A006003 (for natural numbers), A260513 (for triangular numbers), A372583 (for pentagonal numbers), A372751 (for hexagonal numbers), A075664 (for cubes).

Sequence in context: A329109 A044581 A126423 * A186103 A194713 A047638

Adjacent sequences: A072471 A072472 A072473 * A072475 A072476 A072477

Keyword

nonn,easy

Author

Amarnath Murthy, Jun 20 2002

Extensions

Edited by Robert G. Wilson v, Jun 21 2002

Status

approved