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99 is carréphylic - approach of √99=3√11 ~ 9.9498743711

Subsequent approximations of √99 - the position of a fraction indicates whether it is over or under the root-value.
 1 0 1 2 3 4 5 6 7 8 9 10 99 109 119 129 139 149 159 169 179 189 199 1980 2179 2378 2577 2776 2975 3174 3373 3572 3771 3970 39501 ... 0 1 1 1 1 1 1 1 1 1 1 1 10 11 12 13 14 15 16 17 18 19 20 199 219 239 259 279 299 319 339 359 379 399 3970 ...

99 is one less than a square, so the exception mentioned in on root approach applies: 199 and 20, as rendered by the formula, are not the first non-trivial sp-block, but the second, the first being 10 and 1 because 102-99*12 = 1 satisfies the diophantine equation.
 Diophantine equation: s2-99p2 = 1 d = distance to nearest square N2: -1 Smallest non-trivial s: (2*100-1)/1 rational: 199 actual: 199 (10) ⇒ F=398 (20) Smallest non-trivial p: 2*10/1 rational: 20 actual: 20 (1) ⇒ primus foldage=20 (1) v-value tq-blocks: 92-99*12: -18 Number of series: 11

Cross multiplying the red-green pairs renders subsequent solutions of the diophantine equation.
 s 1 10 199 3970 ... p 0 1 20 399 ...

 In the numerator: U(1,10)20 = 1/2*U(2,20)20 - half the secundus of 20. In the denominator: U(0,1)20 = - the primus of 20. as well as ... In the numerator: U(0,99)20 = 99*U(0,1)20 - the 99-fold primus of 20. In the denominator: U(1,10)20 = 1/2*U(2,20)20 - half the secundus of 20. and ... In the numerator: U(-9,9)20 = 9*U(-1,1)20 - the 9-fold quartus of 20. In the denominator: U(1,1)20 = - the tertius of 20.