A396906 - OEIS

%I #17 Jun 13 2026 05:15:11

%S 2,5,15,30,50,65,6,10,15,2,5,0,0,6,8,10,0,27,46,70,85,15,30,50,65,8,

%T 12,20,20,30,30,45,45,60,24,24,36,50,50,65,65,80,80,12,20,30,45,60,4,

%U 2,0,18,16,28,26,26,24,55,50,50,45,10,15,15,33,110,100,400,380,380

%N a(n) = number of face diagonals in Johnson solid J_n.

%H Paolo Xausa, <a href="/A396906/b396906.txt">Table of n, a(n) for n = 1..92</a> (the full sequence).

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

%F a(n) = A396904(n) - A396905(n).

%F a(n) = (1/2)*(Sum_{k=3..A394913(n)} A394912(n,k)*k*(k-3)).

%e a(6) = 65 because Johnson solid J_6 (pentagonal rotunda) has 10 triangular faces (no diagonals), 6 pentagonal faces (5 diagonals each) and 1 10-gonal face (35 diagonals), for a total of 10*0 + 6*5 + 1*35 = 65 face diagonals.

%t Sum[k*(k-3), {k, Map[Length, #]}]/2 & @@@ Array[PolyhedronData[{"Johnson", #}, {"FaceIndices"}] &, 92]

%Y Cf. A394912, A394913, A396904, A396905.

%K nonn,fini,full

%O 1,1

%A _Paolo Xausa_, Jun 09 2026