I'm trying to count up the number of convex polyhedra whose faces are regular polygons and whose largest face is an $n$-gon. (I.e. either a uniform polyhedron or a Johnson solid.) If I've done my counting correct, this does not appear in the On-Line Encyclopedia of Integer Sequences, and I'd like to add it.
Are my lists complete and correct?
For $n = 3$, I've counted $8$:
Tetrahedron,
octahedron,
icosahedron,
$J_{12}$,
$J_{13}$,
$J_{17}$, $J_{51}$,
$J_{84}$.
For $n=4$, I've counted $30$:
Cube,
cuboctahedron,
rhombicuboctahedron,
snub cube,
triangular prism,
square antiprism,
$J_{1}$,
$J_{7}$,
$J_{8}$,
$J_{10}$,
$J_{14}$,
$J_{15}$,
$J_{16}$,
$J_{26}$,
$J_{27}$,
$J_{28}$,
$J_{29}$,
$J_{35}$,
$J_{36}$,
$J_{37}$,
$J_{44}$,
$J_{45}$,
$J_{49}$,
$J_{50}$,
$J_{85}$,
$J_{86}$,
$J_{87}$,
$J_{88}$,
$J_{89}$,
$J_{90}$.
For $n = 5$, I've counted $37$:
Dodecahedron,
icosidodecahedron,
rhombicosidodecahedron,
snub dodecahedron,
pentagonal prism,
pentagonal antiprism,
$J_{2}$,
$J_{9}$,
$J_{11}$,
$J_{30}$,
$J_{31}$,
$J_{38}$,
$J_{39}$,
$J_{46}$,
$J_{52}$,
$J_{53}$,
$J_{62}$,
$J_{63}$,
$J_{64}$,
$J_{91}$,
$J_{32}$,
$J_{33}$,
$J_{40}$,
$J_{41}$,
$J_{47}$,
$J_{61}$,
$J_{59}$,
$J_{60}$,
$J_{58}$,
$J_{34}$,
$J_{42}$,
$J_{43}$,
$J_{48}$,
$J_{72}$,
$J_{73}$,
$J_{74}$,
$J_{75}$.
For $n = 6$, I've counted $14$:
Truncated tetrahedron,
truncated octahedron,
truncated icosahedron,
hexagonal prism,
hexagonal antiprism,
$J_{3}$,
$J_{18}$,
$J_{22}$,
$J_{54}$,
$J_{55}$,
$J_{56}$,
$J_{57}$,
$J_{65}$,
$J_{92}$.
For $n = 7$, I've counted $2$:
Heptagonal prism,
heptagonal antiprism.
For $n = 8$, I've counted $9$:
Truncated cube,
truncated cuboctahedron,
octagonal prism,
octagonal antiprism,
$J_{4}$,
$J_{19}$,
$J_{23}$,
$J_{66}$,
$J_{67}$.
For $n = 9$, I've counted $2$:
Enneagonal prism,
enneagonal antiprism.
For $n = 10$, I've counted $22$:
Truncated dodecahedron,
truncated icosidodecahedron,
decagonal prism,
decagonal antiprism,
$J_{5}$,
$J_{6}$,
$J_{20}$,
$J_{21}$,
$J_{24}$,
$J_{25}$,
$J_{68}$,
$J_{69}$,
$J_{70}$,
$J_{71}$,
$J_{76}$,
$J_{77}$,
$J_{78}$,
$J_{79}$,
$J_{80}$,
$J_{81}$,
$J_{82}$,
$J_{83}$.
For $n > 10$, I've counted $2$:
$n$-gonal prism, $n$-gonal antiprism.
Best Answer
It turns out your list is complete. Having recognized the five Platonic and 13 Archimedean solids by the names you list, I turn to the Johnson solids, which are defined as all other strictly convex (dihedral angles all <180°) polyhedra consisting of regular faces besides prisms and antiprisms. Norman Johnson identified 92 solids, all of which are in your list, in 1966; the numbering we use today is from his identification. That the 92 solids so identified is the complete list of Johnson solids was proved by Victor Zalgaller in 1969.
I also checked your count of the Johnson solids for each individual value of $n$ in your lists. These too are correct.