A250303 Three-column array read by rows: T(n,k) = the coefficient of x^k in the expanded polynomial x^2 + (x+1)^2 + ... + (x+n-1)^2, for 0 <= k <= 2.

0, 0, 1, 1, 2, 2, 5, 6, 3, 14, 12, 4, 30, 20, 5, 55, 30, 6, 91, 42, 7, 140, 56, 8, 204, 72, 9, 285, 90, 10, 385, 110, 11, 506, 132, 12, 650, 156, 13, 819, 182, 14, 1015, 210, 15, 1240, 240, 16, 1496, 272, 17, 1785, 306, 18, 2109, 342, 19, 2470, 380, 20, 2870, 420, 21, 3311, 462, 22, 3795, 506

Offset

1,5

Comments

A001032 solves the Diophantine equation: k^2 + (k+1)^2 + ... + (k+n-1)^2 = y^2. This array gives the coefficients of the left hand side for specified n.

Links

Table of n, a(n) for n=1..68.

Formula

a(3*k+1) = A000330(k), for k >= 0.

a(3*k+2) = A002378(k), for k >= 0.

a(3*k) = k, for k >= 1.

Example

Array starts:
n = 1:   0,  0, 1;
n = 2:   1,  2, 2;
n = 3:   5,  6, 3;
n = 4:  14, 12, 4;
n = 5:  30, 20, 5;
n = 6:  55, 30, 6;
n = 7:  91, 42, 7;
n = 8: 140, 56, 8;
...

Prog

(PARI) for(n=1, 50, for(k=0, 2, print1(polcoeff(sum(i=1, n, (x+i-1)^2), k), ", ")))

Crossrefs

Cf. A000330, A002378, A001032.

Sequence in context: A068763 A176989 A367211 * A368554 A301477 A261895

Adjacent sequences: A250300 A250301 A250302 * A250304 A250305 A250306

Keyword

nonn,easy,tabf

Author

Derek Orr, Jan 15 2015

Status

approved