A218832 Number of positive integer solutions to the Diophantine equation x + y + 2z = n^2.

0, 1, 12, 49, 132, 289, 552, 961, 1560, 2401, 3540, 5041, 6972, 9409, 12432, 16129, 20592, 25921, 32220, 39601, 48180, 58081, 69432, 82369, 97032, 113569, 132132, 152881, 175980, 201601, 229920, 261121, 295392, 332929, 373932, 418609, 467172, 519841, 576840, 638401

Offset: 1

Author

Dennis P. Walsh, Mar 27 2013

Keywords

• nonn
• easy

Comments

The derivation for the number of integer solutions is given in a link below. It is straightforward and uses the fact that the number of positive integer solutions to x + y = n is given by n-1.

Examples

			For n=3, a(n)=12 since there are exactly 12 positive integer solutions (x,y,z) to x+y+2z=9, namely, (1,2,3),(1,4,2), (1,6,1), (2,1,3), (2,3,2), (2,5,1), (3,2,2),(3,4,1), (4,1,2), (4,3,1), (5,2,1), and (6,1,1).
		

Links

• Dennis Walsh, The number of positive integer solutions to the Diophantine equation x+y+2z=n^2

Programs

Maple

seq(floor(n^4/4-n^2+1),n=1..40);

Formula

a(n) = floor(n^4/4-n^2+1).

Conjectures from Colin Barker, Apr 01 2013: (Start)

a(n) = (7 + (-1)^n - 8*n^2 + 2*n^4)/8.

G.f.: -x^2*(x^4 - 4*x^3 + 6*x^2 + 8*x + 1) / ((x-1)^5*(x+1)). (End)