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3 is carréphylic - approach of √3 ~ 1.7320508076

Subsequent approximations of √3 - the position of a fraction indicates whether it is over or under the root-value.
 1 0 1 2 3 5 7 12 19 26 45 71 97 168 265 362 627 989 1351 2340 3691 5042 8733 13775 18817 32592 51409 70226 121635 191861 262087 453948 716035 978122 1694157 ... 0 1 1 1 2 3 4 7 11 15 26 41 56 97 153 209 362 571 780 1351 2131 2911 5042 7953 10864 18817 29681 40545 70226 110771 151316 262087 413403 564719 978122 ...

3 is one less than a square, so the exception mentioned in on root approach applies: 7 and 4, as rendered by the formula, are not the first non-trivial sp-block, but the second, the first being 2 and 1 because 22-3*12 = 1 satisfies the diophantine equation.
 Diophantine equation: s2-3p2 = 1 d = distance to nearest square N2: -1 Smallest non-trivial s: (2*4-1)/1 rational: 7 actual: 7 (2) ⇒ F=14 (4) Smallest non-trivial p: 2*2/1 rational: 4 actual: 4 (1) ⇒ primus foldage=4 (1) v-value qt-blocks: 12-3*12: -2 Number of series: 3

Cross multiplying the red-green pairs renders subsequent solutions of the diophantine equation.
 s 1 2 7 26 97 362 1351 5042 18817 70226 262087 978122 ... p 0 1 4 15 56 209 780 2911 10864 40545 151316 564719 ...

 In the numerator: U(1,2)4 = 1/2*U(2,4)4 - half the secundus of 4. In the denominator: U(0,1)4 = - the primus of 4. as well as ... In the numerator: U(0,3)4 = 3*U(0,1)4 - the 3-fold primus of 4. In the denominator: U(1,2)4 = 1/2*U(2,4)4 - half the secundus of 4. and ... In the numerator: U(-1,1)4 = - the quartus of 4. In the denominator: U(1,1)4 = - the tertius of 4.