Identities in the spirit of Ramanujan's amazing identity

Motivated by an amazing identity by Ramanujan in his "lost notebook", a proof of Ramanujan's identity suggested by Hirschhorn using an algebraic identity, and an algorithm by Chen to find such an algebraic identity, we will establish several identities similar to Ramanujan's amazing identity. For example, if Sigma n >= 0 a(n)x(n) = 9 + 3609x - 135x(2)/1 - 6888x + 6888x(2) - x(3), Sigma n >= 0 b(n)x(n) = 10 - 1478x + 172x(2)/1 - 6888x + 6888x(2) - x(3), Sigma n >= 0 c(n)x(n) = 12 + 1146x + 138x(2)/1 - 6888x + 6888x(2) - x(3), then a(n)(3) + b(n)(3) = c(n)(3) + 1.