cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

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

Views

Author

Clark Kimberling, Apr 27 2000

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A055168, A217760.

Programs

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

Extensions

Corrected and edited by Clark Kimberling, Oct 24 2009
Reconciled to A217760 (formerly A055186) by Clark Kimberling, Mar 25 2013

A055168 Cumulative counting sequence: method B (noun,adjective)-pairs with first term 0.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 1, 0, 3, 1, 3, 2, 1, 0, 4, 1, 5, 2, 2, 3, 2, 0, 5, 1, 6, 2, 5, 3, 3, 4, 1, 5, 1, 0, 6, 1, 9, 2, 6, 3, 5, 4, 2, 5, 4, 6, 1, 0, 7, 1, 11, 2, 8, 3, 6, 4, 4, 5, 6, 6, 4, 9, 1, 0, 8, 1, 13, 2, 9, 3, 7, 4, 7, 5, 7, 6, 7, 9, 2, 7, 1, 11, 1, 8, 1, 0, 9, 1, 17
Offset: 1

Views

Author

Clark Kimberling, Apr 27 2000

Keywords

Comments

Write 0 followed by segments defined inductively as follows: each segment tells how many times each previously written integer occurs, stating first the integer being counted and then its frequency. This is Method B (noun-before-adjective); for Method A (adjective-before-noun), see A217760. - Clark Kimberling, Mar 25 2013

Examples

			Start with 0, then 0,1; then 2,0 and 1,1; etc.
Writing pairs vertically, the initial segments are
0..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
...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 3 1
The 5th segment tells that 0 has been written 4 times, 1 5 times, 2 2 times, and 3 2 times. The nouns are 1 2 3; the adjectives, 5 2 2.  - _Clark Kimberling_, Mar 25 2013

Links

Crossrefs

Cf. A217760, A055170.
Cf. A055171, A055174, A055177, A055180, A055183.
See A001155 and A022470 for well-known counting sequences using methods A and B.

Programs

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

Extensions

Conjecture removed by Clark Kimberling, Oct 24 2009

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

Views

Author

Clark Kimberling, Apr 27 2000

Keywords

Comments

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

Examples

			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; ...

Links

Crossrefs

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

Programs

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

Formula

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

Extensions

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

A217780 Cumulative counting sequence: (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, 3, 3, 2, 4, 1, 6, 1, 2, 11, 1, 5, 3, 3, 4, 2, 6, 3, 2, 1, 8, 13, 1, 8, 3, 4, 4, 3, 6, 5, 2, 2, 8, 1, 11, 1, 5, 16, 1, 10, 3, 6, 4, 4, 6, 7, 2, 4, 8, 2, 11, 3, 5, 1, 13, 18, 1, 12, 3, 9, 4, 6, 6, 9, 2, 5, 8, 3, 11, 4, 5, 2, 13, 1, 16, 1, 10, 1, 7
Offset: 1

Views

Author

Clark Kimberling, Mar 25 2013

Keywords

Comments

After starting with 1, successive segments are generated in adjective-before-noun pairs as in A217760, not as in A055187. That is, noun-integers occur in order of 1st appearance, not increasing order.

Examples

			Writing pairs vertically, the initial segments are
1..1..3..4 1..6 2 1..8 3 2 1 1..11 5 3 2 3 1
...1..1..1 3..1 3 4..1 3 4 6 2...1 3 4 6 2 8.
The 5th segment is read "6 1's and 2 3's and 1 4," this being a count of all the previously written numbers.  The numbers 6,2,1 are used as adjectives, whereas 1,3,4 are used as nouns.  Here, the nouns are kept in order of first appearance; in A055187, they are in increasing order. - _Clark Kimberling_, Mar 25 2013

Links

Crossrefs

Cf. A217760, A055187.

Programs

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

Extensions

More terms from Seiichi Manyama, Nov 10 2022

A055191 Cumulative counting sequence: method A (adjective-before-noun) pairs, starting with 5.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Apr 27 2000

Keywords

Comments

Segments (generated as at A217760): 5; 1,5; 2,5,1,1; 3,5,3,1,1,2; ... The 4th segment is read "3 5's and 3 1's and 1 2" as a record of what was previously written; 3,3,1 are adjectives and 5 1 2 are nouns. Conjecture: every nonnegative integer occurs. - Clark Kimberling, Mar 25 2013

Examples

			Write 5, thus having 1 5, thus having 2 5's and 1 1, thus having 3 5's and 3 1's and 1 2, etc.

Links

Crossrefs

Cf. A217760, A217780.

Programs

  • Mathematica
    s = {5}; Do[s = Flatten[{s, {Count[s, #], #} & /@ DeleteDuplicates[s]}], {14}]; s (* Peter J. C. Moses, Mar 21 2013 *)

A055190 Cumulative counting sequence: method A (adjective-before-noun) pairs, starting with 4.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Apr 27 2000

Keywords

Comments

Segments (as at A217760): 4; 1,4; 2,4,1,1; 3,4,3,1,1,2; ...
The 4th segment is read "3 4's and 3 1's and 1 2" as a record of what was previously written; 3,3,1 are adjectives and 4 1 2 are nouns. Conjecture: every nonnegative integer occurs. - Clark Kimberling, Mar 25 2013

Examples

			Write 4, thus having 1 4, thus having 2 4's and 1 1, thus having 3 4's and 3 1's and 1 2, etc.

Links

Crossrefs

Cf. A217760, A217780.

Programs

  • Mathematica
    s = {4}; Do[s = Flatten[{s, {Count[s, #], #} & /@ DeleteDuplicates[s]}], {14}]; s (* Peter J. C. Moses, Mar 21 2013 *)

A055188 Cumulative counting sequence: method A (adjective-before-noun) pairs, starting with 2.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Apr 27 2000

Keywords

Comments

Segments (generated as at A217760): 2; 1,2; 2,2,1,1; 4,2,3,1; 5,2,4,1,1,4,1,3; ... The 4th segment is read "4 2's and 3 1's" as a record of what was previously written; 4,2 are adjectives and 2 1 are nouns. Conjecture: every nonnegative integer occurs. - Clark Kimberling, Mar 25 2013

Examples

			Write 2, thus having 1 2, thus having 2 2's and 1 1, thus having 4 2's and 3 1's, etc.

Links

Crossrefs

Cf. A217760, A217780.

Programs

  • Mathematica
    s = {2}; Do[s = Flatten[{s, {Count[s, #], #} & /@ DeleteDuplicates[s]}], {14}]; s (* Peter J. C. Moses, Mar 21 2013 *)

A055189 Cumulative counting sequence: method A (adjective-before-noun)-pairs, starting with 3.

Original entry on oeis.org

3, 1, 3, 2, 3, 1, 1, 3, 3, 3, 1, 1, 2, 6, 3, 5, 1, 2, 2, 7, 3, 6, 1, 4, 2, 1, 6, 1, 5, 8, 3, 9, 1, 5, 2, 3, 6, 2, 5, 1, 7, 1, 4, 10, 3, 12, 1, 7, 2, 4, 6, 4, 5, 2, 7, 2, 4, 1, 8, 1, 9, 11, 3, 15, 1, 10, 2, 5, 6, 5, 5, 4, 7, 5, 4, 2, 8, 2, 9, 1, 10, 1, 12, 12, 3, 18
Offset: 1

Views

Author

Clark Kimberling, Apr 27 2000

Keywords

Comments

Segments (generated as at A217760): 3; 1,3; 2,3,1,1; 3,3,3,1,1,2; ...
The 4th segment is read "3 3's and 3 1's and 1 2" as a record of what was previously written; 3 3 1 are adjectives and 3 1 2 are nouns. Conjecture: every nonnegative integer occurs. - Clark Kimberling, Mar 25 2013

Examples

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

Links

Crossrefs

Cf. A217760, A217780.

Programs

  • Mathematica
    s = {3}; Do[s = Flatten[{s, {Count[s, #], #} & /@ DeleteDuplicates[s]}], {14}]; s (* Peter J. C. Moses, Mar 21 2013 *)

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

Views

Author

Robin Powell, Apr 24 2023

Keywords

Comments

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

Examples

			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, ...".

Crossrefs

Cf. A217760, A055186, A342585.

Programs

  • PARI
    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
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.