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The Tertius U(1,1)F
The series around U0:

 U-5 = F5 -F4 -4F3 +3F2 +3F -1 U-4 = F4 -F3 -3F2 +2F +1 U-3 = F3 -F2 -2F +1 U-2 = F2 -F -1 U-1 = F -1 U0 = 1 U1 = 1 U2 = F -1 U3 = F2 -F -1 U4 = F3 -F2 -2F +1 U5 = F4 -F3 -3F2 +2F +1 U6 = F5 -F4 -4F3 +3F2 +3F -1
The tertius is symmetric with regard to U0-U1.
It can be derived from the primus by taking the subsequent differences between Un+1 and Un of that series
(with U0 = u0-u-1 = 1).

 Theorems Theorems have been proved by complete induction. For every integer k: Un | Un+k(2n-1) Basic property Un+1*Un-1 = Un2+(F-2) Theorem1 U2n-1+1 = Un(Un+Un-1) Theorem 2.1 U2n+1 = Un(Un+1+Un) Theorem 2.2 U2n-1-1 = (Un+1-Un-1)(Un-Un-1)/(F-2) Theorem 3.1 U2n-1 = (Un+1-Un-1)(Un+1-Un)/(F-2) Theorem 3.2 Theorems 2 and 3 link terms around Un with terms at twice the index value. I call this the series' development 'from the belly'.

The tertius coefficients matrix
Disregarding signs, this matrix is identical to the quartus coefficients matrix.
The degree of a polynome is one less than its index.
Exponents decrease with steps of 1.
Note that the columns come in pairs with an index shift.
 U1: 1 U2: 1 -1 U3: 1 -1 -1 U4: 1 -1 -2 1 U5: 1 -1 -3 2 1 U6: 1 -1 -4 3 3 -1 U7: 1 -1 -5 4 6 -3 -1 U8: 1 -1 -6 5 10 -6 -4 1 U9: 1 -1 -7 6 15 -10 -10 4 1 U10: 1 -1 -8 7 21 -15 -20 10 5 -1 U11: 1 -1 -9 8 28 -21 -35 20 15 -5 -1 U12: 1 -1 -10 9 36 -28 -56 35 35 -15 -6 1 U13: 1 -1 -11 10 45 -36 -84 56 70 -35 -21 6 1 U14: 1 -1 -12 11 55 -45 -120 84 126 -70 -56 21 7 -1 U15: 1 -1 -13 12 66 -55 -165 120 210 -126 -126 56 28 -7 -1 U16: 1 -1 -14 13 78 -66 -220 165 330 -210 -252 126 84 -28 -8 1 U17: 1 -1 -15 14 91 -78 -286 220 495 -330 -462 252 210 -84 -36 8 1 U18: 1 -1 -16 15 105 -91 -364 286 715 -495 -792 462 462 -210 -120 36 9 -1 U19: 1 -1 -17 16 120 -105 -455 364 1001 -715 -1287 792 924 -462 -330 120 45 -9 -1 U20: 1 -1 -18 17 136 -120 -560 455 1365 -1001 -2002 1287 1716 -924 -792 330 165 -45 -10 1 U21: 1 -1 -19 18 153 -136 -680 560 1820 -1365 -3003 2002 3003 -1716 -1716 792 495 -165 -55 10 1 U22: 1 -1 -20 19 171 -153 -816 680 2380 -1820 -4368 3003 5005 -3003 -3432 1716 1287 -495 -220 55 11 -1 U23: 1 -1 -21 20 190 -171 -969 816 3060 -2380 -6188 4368 8008 -5005 -6435 3432 3003 -1287 -715 220 66 -11 -1 U24: 1 -1 -22 21 210 -190 -1140 969 3876 -3060 -8568 6188 12376 -8008 -11440 6435 6435 -3003 -2002 715 286 -66 -12 1 U25: 1 -1 -23 22 231 -210 -1330 1140 4845 -3876 -11628 8568 18564 -12376 -19448 11440 12870 -6435 -5005 2002 1001 -286 -78 12 1 U26: 1 -1 -24 23 253 -231 -1540 1330 5985 -4845 -15504 11628 27132 -18564 -31824 19448 24310 -12870 -11440 5005 3003 -1001 -364 78 13 -1