1. a) for the first four (smallest) Heegner numbers {1, 2, 3, 7} I came up with the formula:

a(n) = 1+((1 + sqrt(3))^(n-1) - (1 - sqrt(3))^(n-1))/(2*sqrt(3)) for n = 1,2,3,4

b) for the last (largest) four Heegner numbers {19, 43, 67, 163} I came up with the formula:

a(n) = 19+24((1 + sqrt(3))^(n-6) - (1 - sqrt(3))^(n-6))/(2sqrt(3)) for n = 6,7,8,9

c) for the fifth (in the middle) Heegner number {11} I came up with the formula:

a(n) = ((1 + sqrt(3))^n - (1 --sqrt(3))^n)/(2*sqrt(3))/4 for n=5

Note that all 3 above formulas contain an expression of the form:

((1 + sqrt(3))^(m) - (1 - sqrt(3))^(m))

Why?

Is there any way to combine above 3 formulas into one, which would cover all 9 Heegner numbers?

2.

On another note I came up with the following polynomial ratio where denominator is the 7th degree polynomial and numerator is the 2nd degree polynomial:

98577n^7 - 3380698n^6 + 47173986* n^5 - 344092600n^4 + 1400304333n^3 - 3131093242n^2 + 3515708244n - 1438921800)/(60(11449n^2 - 206831*n + 958662))

Above polynomial ratio being essentially of 5th degree gives all 9 Heegner numbers for n=1,2,3,4,5,6,7,8,9

So the question arises: why above 5th degree polynomial yields all 9 Heegner numbers?

Is there a connection between formulas described in 1. and polynomial ratio described in 2. ?

Thanks, Best regards, Alexander R. Povolotsky