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The Secundus U(2,F)F
The series around U0:

 U-5 = F5 -5F3 +5F U-4 = F4 -4F2 +2 U-3 = F3 -3F U-2 = F2 -2 U-1 = F U0 = 2 U1 = F U2 = F2 -2 U3 = F3 -3F U4 = F4 -4F2 +2 U5 = F5 -5F3 +5F
The secundus is symmetric with regard to U0.

It can be derived from the primus by taking the subsequent differences between Un+1 and Un-1 of that series
(with U0 = u1 - u-1 = 2).
 Theorems Theorems have been proved by complete induction. For every integer k: Un | U(2k-1)n Basic property Un+1*Un-1 = Un2+(F+2)(F-2) Theorem1 U2n-1 = Un*Un-1 - F Theorem 2.1 U2n = Un2-2 Theorem 2.2 Theorem 2 links terms around Un with terms at twice the index value. I call this the series' development 'from the belly'.

The secundus coefficients matrix
The degree of a polynome the same as its index.
Exponents decrease with steps of 2.
Note that each column displays the subsequent differences in the next one, or, to put it another way, each column displays the partial sums of the previous one.
 U1: 1 U2: 1 -2 U3: 1 -3 U4: 1 -4 2 U5: 1 -5 5 U6: 1 -6 9 -2 U7: 1 -7 14 -7 U8: 1 -8 20 -16 2 U9: 1 -9 27 -30 9 U10: 1 -10 35 -50 25 -2 U11: 1 -11 44 -77 55 -11 U12: 1 -12 54 -112 105 -36 2 U13: 1 -13 65 -156 182 -91 13 U14: 1 -14 77 -210 294 -196 49 -2 U15: 1 -15 90 -275 450 -378 140 -15 U16: 1 -16 104 -352 660 -672 336 -64 2 U17: 1 -17 119 -442 935 -1122 714 -204 17 U18: 1 -18 135 -546 1287 -1782 1386 -540 81 -2 U19: 1 -19 152 -665 1729 -2717 2508 -1254 285 -19 U20: 1 -20 170 -800 2275 -4004 4290 -2640 825 -100 2 U21: 1 -21 189 -952 2940 -5733 7007 -5148 2079 -385 21 U22: 1 -22 209 -1122 3740 -8008 11011 -9438 4719 -1210 121 -2 U23: 1 -23 230 -1311 4692 -10948 16744 -16445 9867 -3289 506 -23 U24: 1 -24 252 -1520 5814 -14688 24752 -27456 19305 -8008 1716 -144 2 U25: 1 -25 275 -1750 7125 -19380 35700 -44200 35750 -17875 5005 -650 25