A055168 -id:A055168 - cp's OEIS frontend

A055170 n-th distinct number to appear in A055168; also the n-th to appear in A217760.

0, 1, 2, 3, 4, 5, 6, 9, 7, 11, 8, 13, 17, 10, 19, 21, 15, 12, 23, 26, 20, 14, 16, 28, 22, 32, 24, 35, 27, 18, 38, 30, 25, 41, 34, 29, 44, 31, 48, 50, 46, 36, 52, 39, 42, 56, 37, 60, 54, 47, 33, 63, 58, 40, 43, 68, 53, 45, 72, 65, 76, 55, 61, 51

Offset: 1

Clark Kimberling, Apr 27 2000

nonn

Conjecture: this sequence is a permutation of the nonnegative integers.
This is the limiting sequence of the noun-integers in the n-th segment generated as in A217760 (but not A055186); see Example.
The conjecture is true: the number 0 appears in every segment of A055168, and, for n > 0, n appears in the (n+1)-th segment (as the number of occurrences of 0 in the previous segments). - Rémy Sigrist, Oct 16 2017

			Following the adjective-before-noun definition at A217760, the first segments are
0..1..2 1..3 3 1..4 5 2 2..5 6 5 3 1 1..6 9 6 5 2 4 1..
...0..0 1..0 1 2..0 1 2 3..0 1 2 3 4 5..0 1 2 3 4 5 6..
(continuing:)
7 11 8 6 4 6 4 1....8 13 9 7 7 7 5 2 1 1..1
0..1 2 3 4 5 6 9....0..1 2 3 4 5 6 9 7 11 8,
this last segment counting the "8 0's and 13 1's and 9 2's..." which have previously appeared.  The numbers 8, 13, 9 are used as adjectives and the numbers 0 1 2 3 4 5 6 9 7 11 8 (as in A055170) are used as nouns.

Peter J. C. Moses, Table of n, a(n) for n = 1..6000

Cf. A055168, A217760.

s = {0}; Do[s = Flatten[{s, {Count[s, #], #} & /@ (DeleteDuplicates[s])}], {30}]; DeleteDuplicates[s] (* Peter J. C. Moses, Mar 25 2013 *)

Corrected and edited by Clark Kimberling, Oct 24 2009

Reconciled to A217760 (formerly A055186) by Clark Kimberling, Mar 25 2013

A055169 Number of new numbers in n-th segment of A055168; see example line of A055168.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 2, 3, 1, 4, 2, 2, 3, 3, 3, 2, 1, 3, 3, 2, 4, 4, 3, 2, 4, 3, 4, 3, 2, 4, 2, 5, 3, 3, 5, 3, 6, 3, 4, 3, 5, 3, 6, 7, 3, 4, 4, 3, 3, 6, 5, 5, 4, 3, 5, 6, 4, 4, 7, 6, 4, 8, 4, 6, 5, 5, 4, 4, 6, 5, 5, 5, 2, 8, 4, 8, 7, 5, 9, 5, 5, 5, 5

Offset: 1

Clark Kimberling, Apr 27 2000

nonn

Also number of new numbers in n-th segment of A055186.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A055169

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See Links section.
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Definition corrected by Clark Kimberling, Oct 24 2009

More terms from Rémy Sigrist, Oct 17 2017

A055186 Cumulative counting sequence: method A (adjective-before-noun)-pairs with first term 0.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 1, 3, 0, 3, 1, 1, 2, 4, 0, 5, 1, 2, 2, 2, 3, 5, 0, 6, 1, 5, 2, 3, 3, 1, 4, 1, 5, 6, 0, 9, 1, 6, 2, 5, 3, 2, 4, 4, 5, 1, 6, 7, 0, 11, 1, 8, 2, 6, 3, 4, 4, 6, 5, 4, 6, 0, 7, 0, 8, 1, 9, 10, 0, 13, 1, 9, 2, 7, 3, 7, 4, 7, 5, 7, 6, 2, 7, 2, 8, 2, 9, 0, 10, 1, 11, 12, 0, 15, 1, 13, 2, 8, 3, 8, 4

Offset: 1

Clark Kimberling, Apr 27 2000

Start with 0; at n-th step, write down what is in the sequence so far.
"Look and Say" how many times each integer (in increasing order), <= max {existing terms} appears in the sequence. Then concatenate.  Sequence's graph looks roughly like that of A080096.
For the original version, where "increasing order..." is "order of 1st appearance", see A217760.  The conjecture formerly placed here applies to A217760. - Clark Kimberling, Mar 24 2013

			Write 0, thus having 1 0, thus having 2 0's and 1 1, thus having 3 0's and 3 1's and 1 2, etc. 0; 1,0; 2,0,1,1; 3,0,3,1,1,2; ...

Zak Seidov, Table of n, a(n) for n = 1..1019 (the first 22 steps)

Cf. A005150. For other versions see A051120, A079668, A079686.
Cf. A055168-A055185 (method B) and A055187-A055191 (method A).
Cf. A217760.

s={0};Do[ta=Table[{Count[s, # ], # }&/@Range[0,Max[s]]]; s=Flatten[{s,ta}],{22}];s (* Zak Seidov, Oct 23 2009 *)

Conjectures: a(n) < 2*sqrt(n); limit as n goes to infinity Max( a(k) : 1<=k<=n)/sqrt(n) exist = 2. - Benoit Cloitre, Jan 28 2003

Edited by N. J. A. Sloane, Jan 17 2009 at the suggestion of R. J. Mathar
Removed a conjecture. - Clark Kimberling, Oct 24 2009
Entries changed to match b-file. - N. J. A. Sloane, Oct 04 2010

A217760 Cumulative counting sequence: (adjective-before-noun) pairs with first term 0; see Comments.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 1, 3, 0, 3, 1, 1, 2, 4, 0, 5, 1, 2, 2, 2, 3, 5, 0, 6, 1, 5, 2, 3, 3, 1, 4, 1, 5, 6, 0, 9, 1, 6, 2, 5, 3, 2, 4, 4, 5, 1, 6, 7, 0, 11, 1, 8, 2, 6, 3, 4, 4, 6, 5, 4, 6, 1, 9, 8, 0, 13, 1, 9

Offset: 1

Clark Kimberling, Mar 24 2013

Write 0 followed by segments defined inductively as follows: each segment
tells how many times each previously written integer occurs, in the order
of first occurrence.  This is Method A (adjective-before-noun pairs); for
Method B (noun-before-adjective), see A055168.  The sequence A217760 was
originally A055186 (Apr 27 2000); the present A055186 has a shorter
definition and differs from A217760 beginning at the 62nd term.

			Start with 0, followed by the adjective-noun pair 1,0; followed by
adjective-noun pairs 2,0 then 1,1; etc. Writing the pairs vertically,
the initial segments are
0..1..2 1..3 3 1..4 5 2 2..5 6 5 3 1 1..6 9 6 5 2 4 1..7 11 8 6 4 6 4 1
...0..0 1..0 1 2..0 1 2 3..0 1 2 3 4 5..0 1 2 3 4 5 6..0 1  2 3 4 5 6 9
The order of appearance is 0,1,2,3,4,5,6,9,7,11,8,... conjectured at A055170 to include all the nonnegative integers.

Clark Kimberling, Table of n, a(n) for n = 1..9666
Clark Kimberling, Unsolved Problems and Rewards, Problem 4

Cf. A055170, A055186.

s = {0}; Do[s = Flatten[{s, {Count[s, #], #} & /@ DeleteDuplicates[s]}], {14}]; s  (* A217760 *)
s = {0}; Do[s = Flatten[{s, {Count[s, #], #} & /@ (a = DeleteDuplicates[s])}], {24}]; a;  (* A055170 *) (* Peter J. C. Moses, Mar 21 2013 *)

A055174 Cumulative counting sequence: method B (noun,adjective)-pairs with first term 2.

Original entry on oeis.org

2, 2, 1, 2, 2, 1, 1, 2, 4, 1, 3, 2, 5, 1, 4, 4, 1, 3, 1, 2, 6, 1, 7, 4, 3, 3, 2, 5, 1, 2, 8, 1, 9, 4, 4, 3, 4, 5, 2, 6, 1, 7, 1, 2, 10, 1, 12, 4, 7, 3, 5, 5, 3, 6, 2, 7, 2, 8, 1, 9, 1, 2, 13, 1, 15, 4, 8, 3, 7, 5, 5, 6, 3, 7, 4, 8, 2, 9, 2, 10, 1, 12, 1, 2, 16, 1, 18

Offset: 1

Clark Kimberling, Apr 27 2000

nonn

Segments (generated as at A055168): 2; 2,1; 2,2,1,1; 2,4,1,3,2,5; ...

Conjecture: every positive integer occurs.

			Write 2, thus having 2 once, thus having 2 twice and 1 once, thus having 2 4 times and 1 3 times and 2 5 times, etc.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A055168.

s = {2}; Do[s = Flatten[{s, {#,Count[s, #]} & /@ DeleteDuplicates[s]}], {24}]; s (* Peter J. C. Moses, Mar 21 2013 *)

A055171 Cumulative counting sequence: method B (noun,adjective)-pairs with first term 1.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 4, 3, 1, 1, 6, 3, 2, 4, 1, 1, 8, 3, 3, 4, 2, 6, 1, 2, 1, 1, 11, 3, 5, 4, 3, 6, 2, 2, 3, 8, 1, 1, 13, 3, 8, 4, 4, 6, 3, 2, 5, 8, 2, 11, 1, 5, 1, 1, 16, 3, 10, 4, 6, 6, 4, 2, 7, 8, 4, 11, 2, 5, 3, 13, 1, 1, 18, 3, 12, 4, 9, 6, 6, 2, 9, 8, 5, 11, 3, 5

Offset: 1

Clark Kimberling, Apr 27 2000

nonn

Segments (generated as at A055168): 1; 1,1; 1,3; 1,4,3,1; 1,6,3,2,4,1; ...

Conjecture: every positive integer occurs.

			Write 1, thus having 1 once, thus having 1 3 times, thus having 1 4 times and 3 once, etc.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A055168.

s = {1}; Do[s = Flatten[{s, {#,Count[s, #]} & /@ DeleteDuplicates[s]}], {24}]; s (* Peter J. C. Moses, Mar 21 2013 *)

from collections import Counter
from sympy import flatten
t = [1]
[t.extend(flatten(Counter(t).items())) for _ in range(10)]
print(t) # Nicholas Stefan Georgescu, Jan 16 2025

A055177 Cumulative counting sequence: method B (noun,adjective)-pairs with first term 3.

Original entry on oeis.org

3, 3, 1, 3, 2, 1, 1, 3, 3, 1, 3, 2, 1, 3, 6, 1, 5, 2, 2, 3, 7, 1, 6, 2, 4, 6, 1, 5, 1, 3, 8, 1, 9, 2, 5, 6, 3, 5, 2, 7, 1, 4, 1, 3, 10, 1, 12, 2, 7, 6, 4, 5, 4, 7, 2, 4, 2, 8, 1, 9, 1, 3, 11, 1, 15, 2, 10, 6, 5, 5, 5, 7, 4, 4, 5, 8, 2, 9, 2, 10, 1, 12, 1, 3, 12, 1

Offset: 1

Clark Kimberling, Apr 27 2000

nonn

Segments (generated as at A055168): 3; 3,1; 3,2,1,1; 3,3,1,3,2,1; ...

Conjecture: every positive integer occurs.

			Write 3, thus having 3 once, thus having 3 twice and 1 once, thus having 3 3 times and 1 3 times and 2 once, etc.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A055168.

s = {3}; Do[s = Flatten[{s, {#,Count[s, #]} & /@ DeleteDuplicates[s]}], {24}]; s (* Peter J. C. Moses, Mar 21 2013 *)

A055180 Cumulative counting sequence: method B (noun,adjective)-pairs with first term 4.

Original entry on oeis.org

4, 4, 1, 4, 2, 1, 1, 4, 3, 1, 3, 2, 1, 4, 4, 1, 5, 2, 2, 3, 2, 4, 6, 1, 6, 2, 5, 3, 3, 5, 1, 4, 7, 1, 8, 2, 6, 3, 5, 5, 3, 6, 2, 4, 8, 1, 9, 2, 8, 3, 7, 5, 5, 6, 4, 7, 1, 8, 1, 4, 10, 1, 12, 2, 9, 3, 8, 5, 7, 6, 5, 7, 3, 8, 4, 9, 1, 4, 12, 1, 14, 2, 10, 3, 10, 5, 9

Offset: 1

Clark Kimberling, Apr 27 2000

nonn

Segments (generated as at A055168): 4; 4,1; 4,2,1,1; 4,3,1,3,2,1; ...

Conjecture: every positive integer occurs.

			Write 4, thus having 4 once, thus having 4 twice and 1 once, thus having 4 3 times and 1 3 times and 2 once, etc.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A055168.

s = {4}; Do[s = Flatten[{s, {#,Count[s, #]} & /@ DeleteDuplicates[s]}], {24}]; s (* Peter J. C. Moses, Mar 21 2013 *)

A055183 Cumulative counting sequence: method B (noun,adjective)-pairs with first term 5.

Original entry on oeis.org

5, 5, 1, 5, 2, 1, 1, 5, 3, 1, 3, 2, 1, 5, 4, 1, 5, 2, 2, 3, 2, 5, 6, 1, 6, 2, 5, 3, 3, 4, 1, 5, 8, 1, 8, 2, 6, 3, 5, 4, 2, 6, 2, 5, 10, 1, 9, 2, 9, 3, 6, 4, 3, 6, 4, 8, 2, 5, 11, 1, 10, 2, 11, 3, 8, 4, 5, 6, 6, 8, 3, 10, 1, 9, 2, 5, 13, 1, 12, 2, 13, 3, 10, 4, 6, 6

Offset: 1

Clark Kimberling, Apr 27 2000

nonn

Segments (generated as at A055168): 5; 5,1; 5,2,1,1; 5,3,1,3,2,1; ...

Conjecture: every positive integer occurs.

			Write 5, thus having 5 once, thus having 5 twice and 1 once, thus having 5 3 times and 1 3 times and 2 once, etc.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A055168.

s = {5}; Do[s = Flatten[{s, {#,Count[s, #]} & /@ DeleteDuplicates[s]}], {24}]; s (* Peter J. C. Moses, Mar 21 2013 *)

A109973 First repeating NA iterates. The NA (Noun-Adjective) function of a finite sequence s of nonnegative integers is the finite sequence 0a1b2c...mz, where a = #0's in s, b = #1's in s, ..., z = #m's in s, m = greatest term in s.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 2, 0, 1, 1, 3, 2, 1, 3, 3, 0, 1, 1, 3, 2, 1, 3, 3, 0, 1, 1, 3, 2, 2, 3, 3, 4, 1, 0, 1, 1, 4, 2, 2, 3, 2, 4, 2, 5, 1, 0, 1, 1, 5, 2, 2, 3, 2, 4, 1, 5, 2, 6, 1, 0, 1, 1, 5, 2, 4, 3, 1, 4, 1, 5, 1, 6, 2, 7, 1, 0, 1, 1, 6, 2, 4, 3, 1, 4, 1, 5, 1, 6, 1, 7, 2, 8, 1, 0, 1, 1, 7, 2, 4, 3, 1, 4, 1, 5

Offset: 1

Clark Kimberling, Jul 06 2005

nonn

This is a concatenation of finite segments.  The first segment is 01122332, obtained by writing the NA iterates of 0 until repetition occurs: 0, 01, 0111, 0113, 01122031, 02132231, 01122332, after which 01122332 repeats.  It helps to speak your way through: write 0 and say 0 one time - that's 01; then say 0 one time and 1 one time - that's 0111; then say 0 one time and 1 3 times, and so on, until reaching the repeating segment 01122332.  This segment is a fixed point of the NA function.
The second segment is obtained by writing the NA iterates of 1 until repetition occurs: 1, 0011, 0212, 011122, 011322, 01122231, 01132331, 01132133, after which 01132133 repeats, so that the second segment is 01132133.
Third segment (from initial 2): 01132133.
Fourth segment (from initial 3): 0113223341.
Fifth segment (from initial 4): 011422324251.  Here, for the first time, the repetition does not occur immediately after the first occurrence.  Indeed, iteration never reaches a fixed point of the NA function.  Instead, the iterates oscillate between 011422324251 and 011324314251.
These observations prompt questions: (1) what initial segments generate fixed points?  (2) do segments eventually occur periodically, regardless of the choice of initial segment?
- Clark Kimberling, May 08 2011

Cf. A001155, A055168, A191654 (repeating AN iterates).

(* Program computes the NA segment starting with 0 *)
nounAdjective[s_] := Flatten@Transpose@({#1, (Count[s, #1] &) /@ #1} &)[Range[0, Max[s]]];
NestList[nounAdjective[#1] &, nounAdjective[{0}], 7]
(* Next program: the NA segment starting with 1 *)
nounAdjective[s_] := Flatten@Transpose@({#1, (Count[s, #1] &) /@ #1} &)[Range[0, Max[s]]];
NestList[nounAdjective[#1] &, nounAdjective[{1}], 7]
(* ...and so on.  By Peter J. C. Moses, Jun 03 2011 *)

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A055170 n-th distinct number to appear in A055168; also the n-th to appear in A217760.

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A055169 Number of new numbers in n-th segment of A055168; see example line of A055168.

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A055186 Cumulative counting sequence: method A (adjective-before-noun)-pairs with first term 0.

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A217760 Cumulative counting sequence: (adjective-before-noun) pairs with first term 0; see Comments.

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A055174 Cumulative counting sequence: method B (noun,adjective)-pairs with first term 2.

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A055171 Cumulative counting sequence: method B (noun,adjective)-pairs with first term 1.

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A055177 Cumulative counting sequence: method B (noun,adjective)-pairs with first term 3.

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A055180 Cumulative counting sequence: method B (noun,adjective)-pairs with first term 4.

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