A394912 - OEIS

A394912

Irregular triangle read by rows: T(n,k) = number of k-gonal faces in Johnson solid J_n, with 3 <= k <= A394913(n).

%I #25 Apr 16 2026 10:48:39

%S 4,1,5,0,1,4,3,0,1,4,5,0,0,0,1,5,5,1,0,0,0,0,1,10,0,6,0,0,0,0,1,4,3,4,

%T 5,5,5,1,12,1,15,0,1,6,10,6,3,8,4,10,5,16,4,9,0,1,4,13,0,0,0,1,5,15,1,

%U 0,0,0,0,1,10,10,6,0,0,0,0,1,16,3,0,1,20,5,0,0,0,1

%N Irregular triangle read by rows: T(n,k) = number of k-gonal faces in Johnson solid J_n, with 3 <= k <= A394913(n).

%C A 10-gon is the highest-sided polygon a Johnson solid can have as a face.

%C All terms in columns k = 7 and k = 9 are zero (there are no Johnson solids with heptagons or 9-gons as faces).

%H Paolo Xausa, <a href="/A394912/b394912.txt">Table of n, a(n) for n = 1..356</a> (all 92 rows, flattened).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/List_of_Johnson_solids">List of Johnson solids</a>.

%e Triangle begins:

%e n\k| 3 4 5 6 7 8 9 10

%e -------------------------------------

%e 1 | 4, 1;

%e 2 | 5, 0, 1;

%e 3 | 4, 3, 0, 1;

%e 4 | 4, 5, 0, 0, 0, 1;

%e 5 | 5, 5, 1, 0, 0, 0, 0, 1;

%e 6 | 10, 0, 6, 0, 0, 0, 0, 1;

%e 7 | 4, 3;

%e 8 | 4, 5;

%e 9 | 5, 5, 1;

%e 10 | 12, 1;

%e 11 | 15, 0, 1;

%e 12 | 6;

%e 13 | 10;

%e 14 | 6, 3;

%e 15 | 8, 4;

%e 16 | 10, 5;

%e 17 | 16;

%e 18 | 4, 9, 0, 1;

%e 19 | 4, 13, 0, 0, 0, 1;

%e 20 | 5, 15, 1, 0, 0, 0, 0, 1;

%e ...

%e Johnson solid J_6 (pentagonal rotunda) has 10 triangular faces, 0 square faces, 6 pentagonal faces, 0 hexagonal faces, 0 heptagonal faces, 0 octagonal faces, 0 9-gonal faces and 1 10-gonal face.

%t A394912row[n_] := BinCounts[#, {3, Max[#] + 1}] & [Map[Length, PolyhedronData["J" <> ToString[n], "FaceIndices"]]];

%t Array[A394912row, 92] (* Gives the full sequence *)

%o (Python)

%o def A394912(n, k=None): # see A394913 for definition of JS

%o "Return [T(n,k), k = 3 .. A394913(n)] if k is not given."

%o if (L := getattr(A394912, f"J{n}", 0))==0:

%o setattr(A394912, f"J{n}", L := [len(f) for f in JS[n-1]['faces']])

%o return sum(L==k for L in L) if k else [A394912(n, k)for k in range(3,max(L)+1)]

%o for n in range(1,20): print(A394912(n)) # _M. F. Hasler_, Apr 10 2026

%Y Cf. A242731 (row sums), A394913 (row lengths + 2), A394914 (right border).

%Y Cf. A343954 (column k = 3).

%Y Cf. other columns (padded with 0's): A343955 (k = 4), A343956 (k = 5), A343957 (k = 6), A343958 (k = 8), A343959 (k = 10).

%Y Cf. A306949 (number of nonzero elements in each row).

%K nonn,tabf,fini,full

%O 1,1

%A _Paolo Xausa_, Apr 08 2026