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The primus U(0,1)F, the secundus U(2,F)F, the tertius U(1,1)F, and the quartus U(-1,1)F.
All (semi-)symmetric series of the type U(a, b)F are multiples of one of these four.

The Primus U(0,1)F
The series around U0:
 U-5 = -F4 +3F2 -1 U-4 = -F3 +2F U-3 = -F2 +1 U-2 = -F U-1 = -1 U0 = 0 U1 = 1 U2 = F U3 = F2 -1 U4 = F3 -2F U5 = F4 -3F2 +1
The primus is semi-symmetric with regard to U0.

The other three of the fabfour can be derived from it by taking subsequent sums and differences of its terms (quartus and tertius) or subsequent differences between Un+1 and Un-1 (secundus).
Its coefficients matrix is, like the sigma repeated matrix, another representation of Pascal's triangle.

 Theorems Theorems have been proved by complete induction. For every integer k: Un | Ukn Basic property Un+1*Un-1 = (Un+1)(Un-1) Theorem1 One for elegance: the square of a term is 1 more than the product of its flankterms.

 The following theorems link terms around Un with terms at twice the index value. This is the series' development 'from the belly'. Theorem 2 is important as well as elegant: it shows that the primus on odd indices equals (tertius)*(quartus) while on even indices it equals (primus)*(secundus). Thus, factorization of the primus effectively boils down to the factorization of the other three. U2n-1 = (Un-Un-1)(Un+Un-1) Theorem 2.1 U2n = Un(Un+1-Un-1) Theorem 2.2 U2n-1 = Un(Un-Un-2)-1 Theorem 3.1 U2n = (Un+1+Un)(Un-Un-1)-1 Theorem 3.2

The primus coefficients matrix
The degree of a polynome is one less than 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.
Pascal's triangle appears starting at the top and looking right and down: 1 - 11 - 121 - 1331 - 14641 etc. To simplify the comparison, the main diagonal has been indicated.
Like Pascal's triangle, this matrix is the sigma repeated matrix revisited.
 U1: 1 U2: 1 U3: 1 -1 U4: 1 -2 U5: 1 -3 1 U6: 1 -4 3 U7: 1 -5 6 -1 U8: 1 -6 10 -4 U9: 1 -7 15 -10 1 U10: 1 -8 21 -20 5 U11: 1 -9 28 -35 15 -1 U12: 1 -10 36 -56 35 -6 U13: 1 -11 45 -84 70 -21 1 U14: 1 -12 55 -120 126 -56 7 U15: 1 -13 66 -165 210 -126 28 -1 U16: 1 -14 78 -220 330 -252 84 -8 U17: 1 -15 91 -286 495 -462 210 -36 1 U18: 1 -16 105 -364 715 -792 462 -120 9 U19: 1 -17 120 -455 1001 -1287 924 -330 45 -1 U20: 1 -18 136 -560 1365 -2002 1716 -792 165 -10 U21: 1 -19 153 -680 1820 -3003 3003 -1716 495 -55 1 U22: 1 -20 171 -816 2380 -4368 5005 -3432 1287 -220 11 U23: 1 -21 190 -969 3060 -6188 8008 -6435 3003 -715 66 -1 U24: 1 -22 210 -1140 3876 -8568 12376 -11440 6435 -2002 286 -12 U25: 1 -23 231 -1330 4845 -11628 18564 -19448 12870 -5005 1001 -78 1 U26: 1 -24 253 -1540 5985 -15504 27132 -31824 24310 -11440 3003 -364 13

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

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

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