A072474 - OEIS

%I #28 Jun 22 2024 14:09:23

%S 1,13,77,294,855,2071,4403,8492,15189,25585,41041,63218,94107,136059,

%T 191815,264536,357833,475797,623029,804670,1026431,1294623,1616187,

%U 1998724,2450525,2980601,3598713,4315402,5142019,6090755,7174671

%N Sum of next n squares.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

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

%F a(n) = (n/12)*(3n^2+1)*(n^2+2). - _Benoit Cloitre_, Jun 26 2002

%F G.f.: x*(1+3*x+x^2)*(1+4*x+x^2)/(1-x)^6. - _Colin Barker_, Mar 23 2012

%F 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

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

%e 1^2 = 1; 2^2 + 3^2 = 13; 4^2 + 5^2 + 6^2 = 77; ...

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

%o (PARI) for(n=1,50,print1(n*(3*n^2+1)*(n^2+2)/12,","))

%Y Cf. A000290, A072475, A050410.

%Y Cf. A260513 (for triangular numbers), A372583 (for pentagonal numbers), A372751 (for hexagonal numbers), A075664 (for cubes).

%K nonn,easy

%O 1,2

%A _Amarnath Murthy_, Jun 20 2002

%E Edited by _Robert G. Wilson v_, Jun 21 2002