A394912 Irregular triangle read by rows: T(n,k) = number of k-gonal faces in Johnson solid J_n, with 3 <= k <= A394913(n).

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

Offset

1,1

Comments

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

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

Links

Paolo Xausa, Table of n, a(n) for n = 1..356 (all 92 rows, flattened).

Wikipedia, List of Johnson solids.

Example

Triangle begins:
  n\k|   3   4   5   6   7   8   9  10
  -------------------------------------
   1 |   4,  1;
   2 |   5,  0,  1;
   3 |   4,  3,  0,  1;
   4 |   4,  5,  0,  0,  0,  1;
   5 |   5,  5,  1,  0,  0,  0,  0,  1;
   6 |  10,  0,  6,  0,  0,  0,  0,  1;
   7 |   4,  3;
   8 |   4,  5;
   9 |   5,  5,  1;
  10 |  12,  1;
  11 |  15,  0,  1;
  12 |   6;
  13 |  10;
  14 |   6,  3;
  15 |   8,  4;
  16 |  10,  5;
  17 |  16;
  18 |   4,  9,  0,  1;
  19 |   4, 13,  0,  0,  0,  1;
  20 |   5, 15,  1,  0,  0,  0,  0,  1;
  ...
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.

Mathematica

A394912row[n_] := BinCounts[#, {3, Max[#] + 1}] & [Map[Length, PolyhedronData["J" <> ToString[n], "FaceIndices"]]];
Array[A394912row, 92] (* Gives the full sequence *)

Prog

(Python)
def A394912(n, k=None): # see A394913 for definition of JS
    "Return [T(n, k), k = 3 .. A394913(n)] if k is not given."
    if (L := getattr(A394912, f"J{n}", 0))==0:
        setattr(A394912, f"J{n}", L := [len(f) for f in JS[n-1]['faces']])
    return sum(L==k for L in L) if k else [A394912(n, k)for k in range(3, max(L)+1)]
for n in range(1, 20): print(A394912(n)) # M. F. Hasler, Apr 10 2026

Crossrefs

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

Cf. A343954 (column k = 3).

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

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

Sequence in context: A076810 A303144 A370451 * A390489 A397552 A265602

Adjacent sequences: A394909 A394910 A394911 * A394913 A394914 A394915

Keyword

nonn,tabf,fini,full

Author

Paolo Xausa, Apr 08 2026

Status

approved