This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
Triangle starts:
1;
1, 1;
3, 0, 1;
4, 3, 0, 1;
7, 6, 2, 0, 1;
14, 10, 5, 2, 0, 1;
23, 23, 11, 4, 2, 0, 1;
39, 50, 22, 10, 4, 2, 0, 1;
71, 99, 48, 22, 9, 4, 2, 0, 1;
124, 200, 105, 46, 21, 9, 4, 2, 0, 1;
...
In the partition 3+3+2+2+2+1+3+3+1, 2 is repeated consecutively 3 times, no part is repeated consecutively more than 3 times. (3 appears 4 times nonconsecutively.)
b:= proc(n, l, k) option remember; `if`(n=0, 1, add(`if`(
i=l, 0, add(b(n-i*j, i, k), j=1..min(k, n/i))), i=1..n))
end:
T:= (n, k)-> b(n, 0, k)-b(n, 0, k-1):
seq(seq(T(n, k), k=1..n), n=1..14); # Alois P. Heinz, Feb 08 2017
nn=15;Table[Take[Drop[Transpose[Map[PadRight[#,nn+1]&,Table[ CoefficientList[Series[1/(1-Sum[Sum[x^(j i),{i,1,k}]/Sum[x^(j i),{i,0,k}],{j,1,nn}])-1/(1-Sum[Sum[x^(j i),{i,1,k-1}]/Sum[x^(j i),{i,0,k-1}],{j,1,nn}]),{x,0,nn}],x],{k,1,nn}]]],1][[n]],n],{n,1,nn}]//Grid
(* or *)
Needs["Combinatorica`"];Table[Distribution[Map[Max,Map[Length,Map[Split, Level[Map[Permutations,IntegerPartitions[n,n]],{2}]],{2}]],Range[1,n]],{n,1,15}]//Grid (* Geoffrey Critzer, Mar 24 2014 *)
b[n_, l_, k_] := b[n, l, k] = If[n == 0, 1, Sum[If[i == l, 0,
Sum[b[n - i*j, i, k], {j, 1, Min[k, n/i]}]], {i, 1, n}]];
T[n_, k_] := b[n, 0, k] - b[n, 0, k - 1];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 04 2021, after Alois P. Heinz *)
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