(2n-1)^2+(2n+1)^2=((n+1)*(2(n+1)-1)*(2(n+1)+1))/3

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Solution for (2n-1)^2+(2n+1)^2=((n+1)*(2(n+1)-1)*(2(n+1)+1))/3 equation: 


n in (-oo:+oo)

(2*n-1)^2+(2*n+1)^2 = ((n+1)*(2*(n+1)-1)*(2*(n+1)+1))/3 // - ((n+1)*(2*(n+1)-1)*(2*(n+1)+1))/3

(2*n-1)^2+(2*n+1)^2-(((n+1)*(2*(n+1)-1)*(2*(n+1)+1))/3) = 0

(2*n-1)^2+(2*n+1)^2+(-1*(n+1)*(2*(n+1)-1)*(2*(n+1)+1))/3 = 0

(3*(2*n-1)^2)/3+(3*(2*n+1)^2)/3+(-1*(n+1)*(2*(n+1)-1)*(2*(n+1)+1))/3 = 0

3*(2*n-1)^2+3*(2*n+1)^2-1*(n+1)*(2*(n+1)-1)*(2*(n+1)+1) = 0

24*n^2-4*n^3-12*n^2-11*n-3+6 = 0

12*n^2-4*n^3-11*n+3 = 0

12*n^2-4*n^3-11*n+3 = 0

12*n^2-4*n^3-11*n+3 = 0

{ 1, -1, 3, -3 }

1

n = 1

12*n^2-4*n^3-11*n+3 = 0

1

n-1

8*n-4*n^2-3

12*n^2-4*n^3-11*n+3

n-1

4*n^3-4*n^2

8*n^2-11*n+3

8*n-8*n^2

3-3*n

3*n-3

0

8*n-4*n^2-3 = 0

DELTA = 8^2-(-4*(-3)*4)

DELTA = 16

DELTA > 0

n = (16^(1/2)-8)/(-4*2) or n = (-16^(1/2)-8)/(-4*2)

n = 1/2 or n = 3/2

n in { 1/2, 3/2, 1} 

(n-1/2)*(n-3/2)*(n-1) = 0

((n-1/2)*(n-3/2)*(n-1))/3 = 0

((n-1/2)*(n-3/2)*(n-1))/3 = 0 // * 3

(n-1/2)*(n-3/2)*(n-1) = 0

( n-1/2 )

n-1/2 = 0 // + 1/2

n = 1/2

( n-1 )

n-1 = 0 // + 1

n = 1

( n-3/2 )

n-3/2 = 0 // + 3/2

n = 3/2

n in { 1/2, 1, 3/2 }

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