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

Author

Amarnath Murthy, Jun 20 2002

Keywords

• nonn
• easy

Examples

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

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).

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).

Programs

Magma

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

Mathematica

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

PARI

a(n) = n*(3*n^2+1)*(n^2+2)/12

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

Extensions

Edited by Robert G. Wilson v, Jun 21 2002