A240508 -id:A240508 - cp's OEIS frontend

A126027 Row lengths for A333867.

1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 7, 8, 11, 15, 18, 20, 23, 25, 30, 38, 49, 62, 77, 94, 110, 129, 149, 172, 195, 218, 241, 266, 293, 323, 356, 389, 424, 461, 500, 545, 593, 641, 688, 737, 787, 839, 896, 957, 1021, 1085, 1152, 1219, 1291, 1368, 1447, 1527, 1611, 1697

Offset: 1

Franklin T. Adams-Watters, Dec 14 2006

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..1201 (first 500 terms from Reinhard Zumkeller)

Cf. A333867, A240508, A253170.

a126027 = length . a333867_row  -- Reinhard Zumkeller, Dec 28 2014

nn = 69; c[] = 0; k = a[1] = c[1] = 1; Join[{1}, Reap[Do[w = Union@ Array[a, k]; Do[Set[a[j + k], c[w[[j]]]], {j, Length[w]}]; Do[c[a[j + k]]++, {j, Length[w]}]; k += Length[w]; Sow[Max[w]], {n, nn}]][[-1, -1]] ] (* _Michael De Vlieger, Nov 15 2022 *)

a(n+1) = max_{k=1..a(n)} A333867(n,k).

A174382 T(1,0)=0 and for n > 1, T(n,k) is the number of k's in rows 1 to n - 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 4, 0, 1, 2, 6, 0, 1, 1, 3, 8, 1, 1, 1, 0, 1, 4, 12, 1, 2, 1, 0, 1, 0, 1, 6, 16, 2, 2, 2, 0, 1, 0, 1, 0, 0, 0, 1, 11, 19, 5, 2, 2, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 19, 22, 8, 2, 2, 1, 2, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 27, 28, 11, 2, 2, 1, 2, 0, 2, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 1

Offset: 1

Paolo P. Lava & Giorgio Balzarotti, Mar 17 2010

Construction as in A333867 but starting with a 0 and including a count of 0s at the start of each row. [Edited by Peter Munn, Oct 11 2022]

See A342585 for a similarly defined sequence that has been analyzed more and has lists of other related sequences. - Peter Munn, Oct 08 2022

			0;
1; # one zero
1,1; # one zero, one one
1,3; # one zero, three ones
1,4,0,1; # one zero, four ones, zero twos, one three

Reinhard Zumkeller, Rows n = 1..25 of table, flattened

Cf. A240508 (row lengths).

Cf. A333867, A342585.

import Data.List (sort, group)
a174382 n k = a174382_tabf !! (n-1) !! k
a174382_row n = a174382_tabf !! (n-1)
a174382_tabf = iterate f [0] where
   f xs = g (xs ++ [0, 0 ..]) [0..] (map head zs) (map length zs)
     where g   _ [] = []
           g (u:us) (k:ks) hs'@(h:hs) vs'@(v:vs)
             | k == h = u + v : g us ks hs vs
             | k /= h = u : g us ks hs' vs'
           zs = group $ sort xs
-- Reinhard Zumkeller, Apr 06 2014

b:= proc(n) option remember; `if`(n<1, 0, b(n-1)+add(x^i, i=T(n))) end:
T:= proc(n) option remember; `if`(n=1, 0, (p->
      seq(coeff(p, x, i), i=0..degree(p)))(b(n-1)))
    end:
seq(T(n), n=1..12);  # Alois P. Heinz, Aug 25 2025

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A126027 Row lengths for A333867.

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