|
|
A059255
|
|
Both sum of n+1 consecutive squares and sum of the immediately following n consecutive squares.
|
|
10
|
|
|
0, 25, 365, 2030, 7230, 19855, 45955, 94220, 176460, 308085, 508585, 802010, 1217450, 1789515, 2558815, 3572440, 4884440, 6556305, 8657445, 11265670, 14467670, 18359495, 23047035, 28646500, 35284900, 43100525, 52243425, 62875890, 75172930, 89322755
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
In 1879, Dostor gave formulas for all solutions -- see the Dickson link. - Jonathan Sondow, Jun 21 2014
|
|
LINKS
|
|
|
FORMULA
|
a(n) = n*(n + 1)*(2n + 1)*(12n^2 + 12n + 1)/6.
a(0)=0, a(1)=25, a(2)=365, a(3)=2030, a(4)=7230, a(5)=19855, a(n) = 6a(n-1)-15a(n-2)+20a(n-3)-15a(n-4)+6a(n-5)-a(n-6). - Harvey P. Dale, May 09 2011
a(n) = (4*T(n)-n)^2+(4*T(n)-n+1)^2+...+(4*T(n))^2 = (4*T(n)+1)^2+(4*T(n)+2)^2+...+(4*T(n)+n)^2, where T = A000217. See Boardman (2000). - Jonathan Sondow, Mar 07 2013
a(0)=0, a(n) = 25 + 340*C(n-1,1) + 1325*C(n-1,2) + 2210*C(n-1,3) + 1680*C(n-1,4) + 480*C(n-1,5) for n >= 1, where C(a,b) are binomial coefficients. - Kieren MacMillan, Sep 16 2014
|
|
EXAMPLE
|
a(3) = 2030 = 21^2 + 22^2 + 23^2 + 24^2 = 25^2 + 26^2 + 27^2.
|
|
MAPLE
|
|
|
MATHEMATICA
|
Table[1/6(-1+n)(-n+14n^2-36n^3+24n^4), {n, 40}] (* or *) LinearRecurrence[ {6, -15, 20, -15, 6, -1}, {0, 25, 365, 2030, 7230, 19855}, 40] (* Harvey P. Dale, May 09 2011 *)
|
|
PROG
|
(Magma) [n*(n+1)*(2*n+1)*(12*n^2+12*n+1)/6 : n in [0..50]]; // Wesley Ivan Hurt, Jun 21 2014
|
|
CROSSREFS
|
The n+1 consecutive squares start with the square of A014105, while the n consecutive squares start with the square of A001844.
|
|
KEYWORD
|
nice,nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|