A006920 -id:A006920 - cp's OEIS frontend

A030717 The first list after the following procedure: starting with a list [1] and an empty list, repeatedly add the distinct values already in the first list in ascending order to the second list and add the corresponding frequencies of those values to the first list.

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, 1, 10, 11, 18, 4, 2, 3, 3, 5, 1, 1, 1, 1, 14, 12, 20, 5, 3, 3, 3, 5, 1, 2, 2, 1

Offset: 1

Clark Kimberling

The clarifying comment that follows refers to the old name, which was: Row 1, where, at stage k>1, write i in row 1 and j in row 2, where i is the number of j's in row 1, for j=1,2,...,m, where m=max number in row 1 from stages 1 to k-1; state 1 is 1 in row 1.

Numbers j for which the count is 0 are omitted, cf. A333867 for the corresponding sequence where they are included. - Sean A. Irvine, Apr 08 2020

			First list begins:
  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;

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Sean A. Irvine)
Sean A. Irvine, Java program (github)

Cf. A030718 (2nd list), A030719 (row lengths), A006920.
Cf. A030707 (repeatedly adds values and frequencies from both lists).
Cf. A333867 (0 counts included, but not the counts of 0s).
Cf. A030723, A030724, A030725, A030726, A358423, A358424, A358425.

t = {{1}}; Do[AppendTo[t, BinCounts[#, {1, Max[#] + 1}] &[Flatten[t]]], {25}];
DeleteCases[Flatten[t], 0]  (* Peter J. C. Moses, Apr 09 2020 *)

More terms from Franklin T. Adams-Watters, Dec 14 2006
Rolled back to original definition and data by Sean A. Irvine, Apr 08 2020
Name revised in line with A030777 by Peter Munn, Oct 11 2022

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.

Original entry on oeis.org

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

Clark Kimberling

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

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

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)

Cf. A126027 (row lengths), A006920, A030717 (zeros suppressed).
Cf. A174382.
Cf. A253170 (row sums).

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

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 *)

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

A174831 a(n) is the concatenation of the frequency of 0's, 1's, 2's etc. in the entries a(0) up to a(n-1).

Original entry on oeis.org

0, 1, 11, 13, 1401, 26011, 3811101, 4121210101, 61622201010001, 1119522020100010001, 1922822120100110001001, 27281122120200110001002001, 37331622120200210001002001000011

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Mar 30 2010

Like A006920 but starting from a(0)=0.

			a(2) = 1|1 because there is 1 zero and 1 one in the entries a(0) and a(1).
a(3) = 1|3 because there are 1 zero and 3 ones in the entries a(0) to a(2).
a(7) = 4|12|1|2|1|0|1|0|1 because there are 4 zeros, 12 ones, 1 two, 2 threes, 1 four, 0 fives,..., 1 eight in the entries a(0) to a(6).

Index to sequences related to 'say what you see'

Cf. A006920, A177359

Definition corrected by Paolo P. Lava, Apr 04 2010

Definition rephrased, examples added, keyword:base added - R. J. Mathar, Aug 10 2010

cp's OEIS Frontend

A030717 The first list after the following procedure: starting with a list [1] and an empty list, repeatedly add the distinct values already in the first list in ascending order to the second list and add the corresponding frequencies of those values to the first list.

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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.

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A174831 a(n) is the concatenation of the frequency of 0's, 1's, 2's etc. in the entries a(0) up to a(n-1).

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Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions