This program is very simple. It does a lot of work in vain.
Nevertheless the first 1000 terms can be calculated in 5 seconds.
With an advanced program millions of terms are available.

Integer
   n                    // order of the polyiamond
   a, b, c              // side lengths of the polyiamond
   nx = 1000            // number of terms for the sequence
   PS(1 To nx)          // pentagonal polyiamond sequence
   sum = 0              // sum of the first n terms

For c = 1 To nx
  
   For a = c + 1 To nx - c
    
      For b = a - c + 1 To nx                // Type A
         n = 2 * b * c + (a - c) ^ 2
         If n <= nx Then PS(n)++
    
      For b = a + 1 To nx                    // Type B
         n = 2 * (b - a) * c + a * a
         If n <= nx Then PS(n)++
  
   For a = c To nx - c
    
      For b = a + 1 To a + c - 1             // Type C
         n = b * b - 2 * (b - a) * (b - c)
         If n <= nx Then PS(n)++
    
      For b = 1 To nx                        // Type D
         n = 2 * (b + a) * (b + c) - b * b
         If n <= nx Then PS(n)++

For n = 1 To nx
   sum = sum + PS(n)
   Print n, PS(n), sum