A333867 Table with T(1,1) = 1; for n>1, T(n,k) is the number of k's in rows 1 through n-1.

1, 1, 2, 2, 1, 3, 2, 3, 3, 1, 4, 3, 3, 4, 3, 5, 1, 5, 3, 6, 2, 1, 6, 4, 7, 2, 2, 1, 7, 6, 7, 3, 2, 2, 1, 8, 8, 8, 3, 2, 3, 3, 8, 9, 11, 3, 2, 3, 3, 3, 8, 10, 15, 3, 2, 3, 3, 4, 1, 0, 1, 10, 11, 18, 4, 2, 3, 3, 5, 1, 1, 1, 0, 0, 0, 1, 14, 12, 20, 5, 3, 3, 3, 5, 1, 2, 2, 0, 0, 0, 1, 0, 0, 1, 17, 14, 23, 5, 5

Offset

1,3

Comments

Equivalently, list 1, where, at stage k>1, write i in list 1 and j in list 2, where i is the number of j's in list 1, for j=1,2,...,m, where m=max number in list 1 from stages 1 to k-1; stage 1 is 1 in list 1.

Differs from A030717 in that this sequence includes 0's. - Sean A. Irvine, Apr 08 2020

Nevertheless, this sequence starts each row with the count of 1's, not 0's, whose counts are not recorded in the sequence (cf. A174382, which is also initialized with a 0). - Peter Munn, Oct 11 2022

Links

Franklin T. Adams-Watters, First 32 rows, flattened

Michael De Vlieger, Plot a(w(j) + k - 1) at (j,k) for j = 1..60 and w the sequence of partial sums of A126027, showing a(m) = 0 in black, a(m) = 1 in red, and a(m) > 1 in light blue.

Sean A. Irvine, Java program (github)

Example

1;
1;
2;
2, 1;
3, 2;
3, 3, 1;
4, 3, 3;

Mathematica

t = {{1}}; Do[AppendTo[t, BinCounts[#, {1, Max[#] + 1}] &[Flatten[t]]], {30}];
Map[Length, t] (* A126027*)
Map[Total, t]  (* A253170*)
Flatten[t]     (* A333867 *)  (* Peter J. C. Moses, Apr 09 2020 *)

Prog

(Haskell)
import Data.List (sort, group)
a030717 n k = a030717_tabf !! (n-1) !! (k-1)
a030717_row n = a030717_tabf !! (n-1)
a030717_tabf = [1] : f [1] where
   f xs = ys : f ((filter (> 0) ys) ++ xs) where
          ys = h (group $ sort xs) [1..] where
               h [] _ = []
               h vss'@(vs:vss) (w:ws)
                 | head vs == w = (length vs) : h vss ws
                 | otherwise    = 0 : h vss' ws
-- Reinhard Zumkeller, Dec 28 2014

Crossrefs

Cf. A126027 (row lengths), A006920, A030717 (zeros suppressed).

Cf. A174382.

Cf. A253170 (row sums).

Sequence in context: A322529 A349526 A329949 * A030717 A280716 A319444

Adjacent sequences: A333864 A333865 A333866 * A333868 A333869 A333870

Keyword

nonn,tabf,nice

Author

Clark Kimberling

Extensions

More terms from Franklin T. Adams-Watters, Dec 14 2006

Status

approved