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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