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The Quartus 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 quartus is symmetric with regard to U0-U1.
It can be derived from the primus by taking the subsequent sums of 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+1-Un-1)(Un-Un-1+Un-2- ... ±1) Theorem 2.2 U2n-1-1 = (Un+1-Un-1)(Un-1-Un-2+Un-3- ... ±1) Theorem 3.1 U2n-1 = Un(Un+1-Un) 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'. Note: The quartus of the factor (F2-2) is the term by term product of the tertius and the quartus of the factor F, which is also the primus of F on odd indices (primus theorem 2.1).

The quartus coefficients matrix
Disregarding signs, this matrix is identical to the tertius 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 -716 -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