A055186 -id:A055186 - cp's OEIS frontend

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

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 *)

A055187 Cumulative counting sequence: method A (adjective-before-noun)-pairs with first term 1.

Original entry on oeis.org

1, 1, 1, 3, 1, 4, 1, 1, 3, 6, 1, 2, 3, 1, 4, 8, 1, 1, 2, 3, 3, 2, 4, 1, 6, 11, 1, 3, 2, 5, 3, 3, 4, 2, 6, 1, 8, 13, 1, 5, 2, 8, 3, 4, 4, 1, 5, 3, 6, 2, 8, 1, 11, 16, 1, 7, 2, 10, 3, 6, 4, 3, 5, 4, 6, 4, 8, 2, 11, 1, 13, 18, 1, 9, 2, 12, 3, 9, 4, 4, 5, 6, 6, 1, 7, 5, 8, 1, 10, 3, 11, 2, 13, 1, 16, 22, 1

Offset: 1

Clark Kimberling, Apr 27 2000

nonn

Conjectures: limit as n goes to infinity of max {a(k) : 1<=k<=n}/sqrt(n) = 2;
-3 < a(n) - 2*sqrt(n) < 3 for all n;
there are infinitely many n such that a(n)=a(n+1). - Benoit Cloitre, Jan 30 2003
After starting with 1, successive segments are generated in adjective-before-noun pairs as in A055186 (i.e., the noun-integers are in increasing order).  See A217780 for the sequence originally placed here, in which the noun-integers are in order of 1st occurrence. - Clark Kimberling, Mar 24 2013

			After writing 1, pairs, written vertically, are as shown:
1..1..3..4 1..6 2 1..8 1 3 2 1..
...1..1..1 3..1 3 4..1 2 3 4 6..

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

Cf. A055186, A217780.

s = {1}; Do[s = Flatten[{s, {Count[s, #], #} & /@ Union[s]}], {14}]; s  (* A055187 *) (* Peter J. C. Moses, Mar 21 2013 *)

Corrected and extended by Benoit Cloitre, Jan 30 2003

A051120 Start with 1; at n-th step, write down what is in the sequence so far.

Original entry on oeis.org

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

Offset: 0

Jamie (sunshinebaby(AT)hotmail.com)

			After 1 1 1 3 1, we see "1 3 and 4 1's", so next terms are 1 3 4 1. Then "1 4, 2 3's, 6 1's"; etc.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A005150, A079668, A055186, A079686.

import Data.List (sort, nub, group)
a051120 n = a051120_list !! n
a051120_list = 1 : f [1] where
  f xs = seen ++ (f $ xs ++ seen) where
    seen = look (reverse $ map length $ group xs') (reverse $ nub xs')
    xs' = sort xs
    look [] []               = []
    look (cnt:cnts) (nr:nrs) = cnt : nr : look cnts nrs
-- Reinhard Zumkeller, Jun 22 2011

s={1}; Do[s=Flatten[{s,{Count[s,#],#}&/@Reverse[Union[s]]}], {60}]; s (* Peter J. C. Moses, Mar 21 2013 *)

More terms from Michael Lugo (mlugo(AT)thelabelguy.com), Dec 22 1999

a(28) corrected by Reinhard Zumkeller, Jun 22 2011

A079603 "Trim" numbers: see reference for definition.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 147, 149, 151, 157, 163, 167, 173, 179, 181, 189, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263

Offset: 1

N. J. A. Sloane, Sep 22 2008

The old entry with this sequence number was a duplicate of A055186.

Ray Chandler, Table of n, a(n) for n = 1..10000
Bhupinder Singh Anand, A Minimal Prime Generating Theorem that suggests the Prime Difference is O(pi(p(n)^(1/2)))
Sean A. Irvine, Java program (github)

Cf. A070566.

Cf. A145555.

Extended by Ray Chandler, Oct 13 2008

A079668 Start with 1; at n-th step, write down what is in the sequence so far.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane Jan 26 2003

After 1 1 1 3 1, we see "4 1's, 0 2's and 1 3", so next terms are 4 1 0 2 3 1.

			Row n lists all terms written at the n-th step:
  1;
  1, 1;
  3, 1;
  4, 1,  0, 2, 1, 3;
  1, 0,  6, 1, 1, 2, 2, 3, 1, 4;
  2, 0, 10, 1, 3, 2, 3, 3, 2, 4, 0, 5, 1, 6;
  4, 0, 12, 1, 6, 2, 6, 3, 3, 4, 1, 5, 2, 6, 0, 7, 0, 8, 0, 9, 1, 10;
  ...

Alois P. Heinz, Table of n, a(n) for n = 1..20377 (first 55 steps)

For other versions see A051120, A055186, A079686.

Cf. A030717, A333867.

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, 1, (p->
      seq([coeff(p, x, i), i][], i=ldegree(p)..degree(p)))(b(n-1)))
    end:
seq(T(n), n=1..10);  # Alois P. Heinz, Aug 24 2025

s={1}; Do[s=Flatten[{s,{Count[s,#],#}&/@Range[Min[s],Max[s]]}],{20}];s (* Peter J. C. Moses, Mar 21 2013 *)

from itertools import islice
def agen(): # generator of terms
    a = [1]
    yield 1
    while True:
        counts = []
        for d in range(min(a), max(a)+1):
            c = a.count(d)
            counts.extend([c, d])
        a += counts
        yield from counts
print(list(islice(agen(), 84))) # Michael S. Branicky, Aug 24 2025

More terms from Sean A. Irvine, Aug 24 2025

A079686 Start with 0; at n-th step, write down what is in the sequence so far.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane Jan 26 2003

Start with the highest value currently in the sequence and count down to zero.

			After the sequence is "0, 1, 0, 1, 1, 2, 0", we see one "2", three "1"s, and three "0"s, so we add 1, 2, 3, 1, 3, 0 to the sequence.

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Cf. A005150. For other versions see A051120, A079668, A055186.

a=[0];c=2;d=2; for j=1:7 for k=max(a):-1:0 a(c)=size(find(a(1:d-1)==k),2); a(c+1)=k; c=c+2; end; d=c; end; a' % Nathaniel Johnston, Apr 25 2011

s={0}; Do[s=Flatten[{s, {Count[s,#],#}&/@Range[Max[s],0,-1]}], {60}]; s (* Peter J. C. Moses, Mar 21 2013 *)

a(13) - a(75) from Nathaniel Johnston, Apr 25 2011

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

Original entry on oeis.org

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

PARI
```
See Links section.
```

Definition corrected by Clark Kimberling, Oct 24 2009

More terms from Rémy Sigrist, Oct 17 2017

A362557 Start with first term 0, then add paired terms counting every preceding term up to the largest term so far and loop back to 0 after every pair has been counted.

Original entry on oeis.org

0, 1, 0, 1, 1, 2, 0, 3, 1, 1, 2, 1, 3, 3, 0, 6, 1, 2, 2, 3, 3, 1, 6, 4, 0, 8, 1, 4, 2, 5, 3, 2, 4, 1, 5, 2, 6, 1, 8, 5, 0, 11, 1, 7, 2, 6, 3, 3, 4, 3, 5, 4, 6, 1, 7, 2, 8, 1, 11, 6, 0, 14, 1, 9, 2, 9, 3, 5, 4, 5, 5, 6, 6, 2, 7, 3, 8, 2, 9, 2, 11, 1, 14, 7, 0

Offset: 1

Robin Powell, Apr 24 2023

Same as A055186, except previous pairs from the same row are included in the count.

			Write "0". There is now "1 0". Now there is "1 1". We can't find any terms greater than 1, so we recheck the sequence for 0s and find "2 0(s)". Listing these terms in the order read out loud yields the sequence "0, 1, 0, 1, 1, 2, 0, ...".

Cf. A217760, A055186, A342585.

seq(n)={my(L=List([0]), m=0, k=0); while(#Lt==k, L)); if(c, listput(L,c); listput(L,k); m=max(m,c));  k=if(k==m, 0, k+1)); Vec(L)} \\ Andrew Howroyd, May 02 2023

cp's OEIS Frontend

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

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A055187 Cumulative counting sequence: method A (adjective-before-noun)-pairs with first term 1.

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A051120 Start with 1; at n-th step, write down what is in the sequence so far.

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Haskell

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A079603 "Trim" numbers: see reference for definition.

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A079668 Start with 1; at n-th step, write down what is in the sequence so far.

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Maple

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A079686 Start with 0; at n-th step, write down what is in the sequence so far.

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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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