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
U5:1-31
U6:1-43
U7:1-56-1
U8:1-610-4
U9:1-715-101
U10:1-821-205
U11:1-928-3515-1
U12:1-1036-5635-6
U13:1-1145-8470-211
U14:1-1255-120126-567
U15:1-1366-165210-12628-1
U16:1-1478-220330-25284-8
U17:1-1591-286495-462210-361
U18:1-16105-364715-792462-1209
U19:1-17120-4551001-1287924-33045-1
U20:1-18136-5601365-20021716-792165-10
U21:1-19153-6801820-30033003-1716495-551
U22:1-20171-8162380-43685005-34321287-22011
U23:1-21190-9693060-61888008-64353003-71566-1
U24:1-22210-11403876-856812376-114406435-2002286-12
U25:1-23231-13304845-1162818564-1944812870-50051001-781
U26:1-24253-15405985-1550427132-3182424310-114403003-36413



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 - FTheorem 2.1
U2n =Un2-2Theorem 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
U4:1-42
U5:1-55
U6:1-69-2
U7:1-714-7
U8:1-820-162
U9:1-927-309
U10:1-1035-5025-2
U11:1-1144-7755-11
U12:1-1254-112105-362
U13:1-1365-156182-9113
U14:1-1477-210294-19649-2
U15:1-1590-275450-378140-15
U16:1-16104-352660-672336-642
U17:1-17119-442935-1122714-20417
U18:1-18135-5461287-17821386-54081-2
U19:1-19152-6651729-27172508-1254285-19
U20:1-20170-8002275-40044290-2640825-1002
U21:1-21189-9522940-57337007-51482079-38521
U22:1-22209-11223740-800811011-94384719-1210121-2
U23:1-23230-13114692-1094816744-164459867-3289506-23
U24:1-24252-15205814-1468824752-2745619305-80081716-1442
U25:1-25275-17507125-1938035700-4420035750-178755005-65025



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
U4:1-1-21
U5:1-1-321
U6:1-1-433-1
U7:1-1-546-3-1
U8:1-1-6510-6-41
U9:1-1-7615-10-1041
U10:1-1-8721-15-20105-1
U11:1-1-9828-21-352015-5-1
U12:1-1-10936-28-563535-15-61
U13:1-1-111045-36-845670-35-2161
U14:1-1-121155-45-12084126-70-56217-1
U15:1-1-131266-55-165120210-126-1265628-7-1
U16:1-1-141378-66-220165330-210-25212684-28-81
U17:1-1-151491-78-286220495-330-462252210-84-3681
U18:1-1-1615105-91-364286715-495-792462462-210-120369-1
U19:1-1-1716120-105-4553641001-715-1287792924-462-33012045-9-1
U20:1-1-1817136-120-5604551365-1001-200212871716-924-792330165-45-101
U21:1-1-1918153-136-6805601820-1365-300320023003-1716-1716792495-165-55101
U22:1-1-2019171-153-8166802380-1820-436830035005-3003-343217161287-495-2205511-1
U23:1-1-2120190-171-9698163060-2380-618843688008-5005-643534323003-1287-71522066-11-1
U24:1-1-2221210-190-11409693876-3060-8568618812376-8008-1144064356435-3003-2002715286-66-121
U25:1-1-2322231-210-133011404845-3876-11628856818564-12376-194481144012870-6435-500520021001-286-78121
U26:1-1-2423253-231-154013305985-4845-155041162827132-18564-318241944824310-12870-1144050053003-1001-3647813-1



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
U2:11
U3:11-1
U4:11-2-1
U5:11-3-21
U6:11-4-331
U7:11-5-463-1
U8:11-6-5106-4-1
U9:11-7-61510-10-41
U10:11-8-72115-20-1051
U11:11-9-82821-35-20155-1
U12:11-10-93628-56-353515-6-1
U13:11-11-104536-84-567035-21-61
U14:11-12-115545-120-8412670-56-2171
U15:11-13-126655-165-120210126-126-56287-1
U16:11-14-137866-220-165330210-252-1268428-8-1
U17:11-15-149178-286-220495330-462-25221084-36-81
U18:11-16-1510591-364-286715495-792-462462210-120-3691
U19:11-17-16120105-455-3641001715-1287-792924462-330-120459-1
U20:11-18-17136120-560-45513651001-2002-12871716924-792-33016545-10-1
U21:11-19-18153136-680-56018201365-3003-20023003-716-1716-792495165-55-101
U22:11-20-19171153-816-68023801820-4368-300350053003-3432-17161287495-220-55111
U23:11-21-20190171-969-81630602380-6188-436880085005-6435-343230031287-715-2206611-1
U24:11-22-21210190-1140-96938763060-8568-6188123768008-11440-643564353003-2002-71528666-12-1
U25:11-23-22231210-1330-114048453876-11628-85681856412376-19448-11440128706435-5005-20021001286-78-121
U26:11-24-23253231-1540-133059854845-15504-116282713218564-31824-194482431012870-11440-500530031001-364-78131