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 1-8 of 8 results.

A003513 Number of regular sequences of length n.

Original entry on oeis.org

1, 2, 6, 27, 192, 2280, 47097, 1735803, 115867758, 14137353466, 3172486137982, 1315460211433262, 1011773137731861712, 1448486351628212391462, 3872217739919424676743213
Offset: 2

Views

Author

N. J. A. Sloane

Keywords

Comments

From Nathaniel Johnston, Jun 29 2023: (Start)
A sequence x_1, ..., x_n is regular if 1 = x_1 <= x_2 <= ... <= x_n and x_j <= Sum_{i=1..j-1} x_i for all j >= 2. It is immediate from this definition that x_2 = 1 and x_j <= 2^(j-2) for all j >= 2.
A sequence x_1, x_2, ..., x_n is regular if and only if (x_2, ..., x_n) is a complete partition of x_2+...+x_n (see A126796 for the definition of a complete partition). As a result, the number of regular sequences with sum equal to n is given by A126796(n-1).
(End)

Examples

			From _Nathaniel Johnston_, Jun 29 2023: (Start)
When n = 4, there are 6 regular sequences:
1,1,1,1
1,1,1,2
1,1,1,3
1,1,2,2
1,1,2,3
1,1,2,4
(End)

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

  • Marc Davio, Unpublished notes, 1975, from a letter to N. J. A. Sloane sent in May 1975.
  • Peter C. Fishburn and Fred S. Roberts, Uniqueness in finite measurement, Applications of combinatorics and graph theory to the biological and social sciences, 103--137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099)
  • Peter C. Fishburn and Fred S. Roberts, Uniqueness in finite measurement, in Applications of combinatorics and graph theory to the biological and social sciences, 103--137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099). [Annotated scan of five pages only]
  • Peter C. Fishburn et al., Van Lier Sequences, Discrete Appl. Math. 27 (1990), pp. 209-220.
  • Nathaniel Johnston and Sarah Plosker, Laplacian {-1,0,1}- and {-1,1}-diagonalizable graphs, arXiv:2308.15611 [math.CO], 2023.

Crossrefs

Sequences in the Fishburn-Roberts (1989) article: A005269, A005268, A234595, A005272, A003513, A008926.
Cf. A126796.

Programs

  • Maple
    A003513 := proc() local a,b,n ; a := {[1,1]} ; n := 3 ; while true do b := {} ; for s in a do subsa := combinat[choose](s) ; for i in subsa do newa := add(k,k=i) ; if newa >= op(-1,s) then b := b union {[op(s),newa]} ; fi ; od; od; print(n,nops(b) ) ; a := b ; n := n+1 ; od; end: A003513() ; # R. J. Mathar, Oct 22 2007

Extensions

a(9) from R. J. Mathar, Oct 22 2007
a(10) from Sean A. Irvine, Jun 15 2015
a(11)-a(16) from Bert Dobbelaere, Dec 28 2020

A005268 Number of elementary sequences of length n.

Original entry on oeis.org

1, 1, 2, 4, 10, 31, 120, 578, 3422, 24504, 208833, 2086777, 24123293, 318800755, 4766262421, 79874304340, 1488227986802
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

In Fishburn-Roberts (1989) it is stated that no recurrence is known. - N. J. A. Sloane, Jan 04 2014

References

  • Fishburn, Peter C.; Roberts, Fred S., Uniqueness in finite measurement. Applications of combinatorics and graph theory to the biological and social sciences, 103--137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099)
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Sequences in the Fishburn-Roberts (1989) article: A005269, A005268, A234595, A005272, A003513, A008926.

Extensions

a(11) corrected and a(12)-a(14) from Sean A. Irvine, Apr 27 2016
a(15)-a(17) from Bert Dobbelaere, Dec 28 2020

A005272 Number of Van Lier sequences of length n.

Original entry on oeis.org

1, 2, 6, 26, 164, 1529, 21439, 461481, 15616226, 851607867, 76555549499, 11550559504086
Offset: 2

Views

Author

N. J. A. Sloane

Keywords

Comments

From Fishburn et al.'s abstract (from the 1990 article): "We study two types of sequences of positive integers which arise from problems in the measurement of comparative judgements of probability. The first type consists of the Van Lier sequences, which are nondecreasing sequences x_1, x_2, ..., x_n of positive integers that start with two 1's and have the property that, whenever j < k <= n, x_k - x_j can be expressed as a sum of terms from the sequence other than x_j. The second type consists of the regular sequences, which are nondecreasing sequences of positive integers that start with two 1's and have the property that each subsequent term is a partial sum of preceding terms. ... We also study one-term extensions of Van Lier sequences and obtain some asymptotic results on the number of Van Lier sequences." - Jonathan Vos Post, Apr 16 2011

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

  • Peter C. Fishburn, Fred S. Roberts, Uniqueness in finite measurement, Applications of combinatorics and graph theory to the biological and social sciences, 103-137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099)
  • Peter C. Fishburn, Fred S. Roberts, Uniqueness in finite measurement, in Applications of combinatorics and graph theory to the biological and social sciences, 103-137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099). [Annotated scan of five pages only]
  • P. C. Fishburn et al., Van Lier Sequences, Discrete Appl. Math. 27 (1990), pp. 209-220.

Crossrefs

Sequences in the Fishburn-Roberts (1989) article: A005269, A005268, A234595, A005272, A003513, A008926.

Extensions

a(9)-a(10) from Sean A. Irvine, Apr 29 2016
a(11)-a(13) from Bert Dobbelaere, Jan 08 2020

A008926 Number of uniquely agreeing sequences.

Original entry on oeis.org

1, 1, 2, 8, 102
Offset: 1

Views

Author

N. J. A. Sloane, Mauro Torelli (torelli(AT)hermes.mc.dsi.unimi.it)

Keywords

References

  • Fishburn, Peter C.; Roberts, Fred S., Uniqueness in finite measurement. Applications of combinatorics and graph theory to the biological and social sciences, 103--137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099)

Links

  • Fishburn, Peter C.; Roberts, Fred S., Uniqueness in finite measurement, in Applications of combinatorics and graph theory to the biological and social sciences, 103--137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099). [Annotated scan of five pages only]
  • P. C. Fishburn et al., Van Lier Sequences, Discrete Appl. Math. 27 (1990), pp. 209-220.

Crossrefs

Sequences in the Fishburn-Roberts (1989) article: A005269, A005268, A234595, A005272, A003513, A008926.

A234595 Number of elementary sequences of length n, where permutations of the components are taken into account.

Original entry on oeis.org

1, 1, 4, 23, 256, 4647, 128262, 5128503
Offset: 1

Views

Author

N. J. A. Sloane, Jan 04 2014

Keywords

References

  • Fishburn, Peter C.; Roberts, Fred S., Uniqueness in finite measurement. Applications of combinatorics and graph theory to the biological and social sciences, 103--137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099)

Links

  • Fishburn, Peter C.; Roberts, Fred S., Uniqueness in finite measurement, in Applications of combinatorics and graph theory to the biological and social sciences, 103--137, IMA Vol. Math. Appl., 17, Springer, New York, 1989. MR1009374 (90e:92099). [Annotated scan of five pages only]

Crossrefs

Sequences in the Fishburn-Roberts (1989) article: A005269, A005268, A234595, A005272, A003513, A008926.

A005270 Number of sequences s of length n with s[1]=1, s[2]=1, s[j-1]=3.

Original entry on oeis.org

1, 1, 1, 2, 6, 27, 177, 1680, 23009, 455368, 13067353, 546378617, 33472296082, 3021920660821, 404374532614122, 80646410554881100, 24095492607316134304, 10837141045948365696938, 7369252748590790186483284, 7606603491185739308318700818
Offset: 2

Views

Author

N. J. A. Sloane

Keywords

Comments

The sequences of length n that are counted here are sub-Fibonacci sequences (A005269) with the property that its members, except for the initial two terms, strictly increase. - Emeric Deutsch, Feb 15 2007

Examples

			G.f. = x^2 + x^3 + x^4 + 2*x^5 + 6*x^6 + 27*x^7 + 177*x^8 + 1680^x^9 + ...
a(2)=6 because we have (1,1,2,3,4,5), (1,1,2,3,4,6), (1,1,2,3,4,7), (1,1,2,3,5,6), (1,1,2,3,5,7) and (1,1,2,3,5,8).

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A125051, A125052.
Cf. A005269.

Programs

  • Maple
    g[0]:=1:for k from 0 to 20 do g[k+1]:=expand(sum(subs({x=y, y=z}, g[k]), z=y+1..x+y)) od:seq(subs({x=1, y=1}, g[k]), k=0..20); # Emeric Deutsch, Feb 15 2007
  • PARI
    {a(n) = if(n<2, return(0)); my(c, e); forvec(s=vector(n, i, [1, fibonacci(i)]), e=0; for(k=3, n, if( s[k-1]>=s[k] || s[k]>s[k-2]+s[k-1], e=1; break)); if(e, next); c++, 1); c}; /* Michael Somos, Dec 02 2016 */

Formula

a(n) equals the number of nodes in generation n-2 of the sub-Fibonacci tree (A125051) for n>=2. - Paul D. Hanna, Nov 19 2006
See the Maple program; g[k](x, y) is the number of sequences s[1], s[2], ..., s[k+2] such that s[1]=x, s[2]=y, s[j-1] =3. - Emeric Deutsch, Feb 15 2007

Extensions

a(12) from Paul D. Hanna, Nov 19 2006
Edited by Emeric Deutsch, Feb 15 2007

A128094 Number of sequences s of length n, with s[1]=1, s[2]=1, s[3]=1, s[k-1] <=s[k] <= s[k-1]+s[k-2]+s[k-3] (s is called a sub-tribonacci sequence of length n).

Original entry on oeis.org

1, 3, 9, 36, 228, 2196, 33901, 862503, 36346723, 2564238411, 304902857694, 61384367733677, 21020435566780278, 12292402317454051941, 12319906894146608845054, 21234027294331775378957366
Offset: 3

Views

Author

Emeric Deutsch, Feb 14 2007

Keywords

Examples

			a(5)=9 because we have (1,1,1,1,1), (1,1,1,1,2), (1,1,1,1,3), (1,1,1,2,2), (1,1,1,2,3), (1,1,1,2,4), (1,1,1,3,3), (1,1,1,3,4), (1,1,1,3,5).

Links

Crossrefs

Cf. A005269.

Programs

  • Maple
    f[0]:=1:for k from 0 to 20 do f[k+1]:=factor(sum(subs({x=y, y=z, z=u}, f[k]), u=z..x+y+z)) od: seq(subs({x=1, y=1,z=1}, f[k]), k=0..20);

Formula

See the Maple program; f[k](x,y,z) is the number of sequences s[1], s[2], ..., s[k+3] such that s[1]=x, s[2]=y, s[3]=z, s[j-1] <=s[j] <= s[j-3]+s[j-2]+s[j-1].

A355129 a(n) is the number of integer sequences b(0..n) of length n+1, with 0 <= b(k) <= k! and monotonic b(k) <= b(k+1).

Original entry on oeis.org

2, 3, 7, 40, 856, 91821, 60080136, 279276911843, 10503211888973754, 3585680755683196123365, 12323227994417456429490342865, 468378989392773003347310901356953089, 214565221409985003242070442557341938941878313, 1282499669290042152350268651085002913530161723080398635
Offset: 0

Views

Author

Thomas Scheuerle, Aug 04 2022

Keywords

Comments

List of the possible cases regarding the patterns of the numbers in the sequence b:
Length: 1 2 3 4 5 6
Pos 0: 1 1 1 1 1 1
Pos 1: 1 2 3 4 5 6
Pos 2: 0 0 3 7 12 18
Pos 3: 0 0 0 7 19 37
Pos 4: 0 0 0 7 26 63
Pos 5: 0 0 0 7 33 96
Pos 6: 0 0 0 7 40 136
Pos 7: 0 0 0 0 40 176
Pos 8: 0 0 0 0 40 216
... ... ... ... ... ... ...
Sum: 2 3 7 40 856 91821
Each row counts the number of possible distributions of numbers, row "Pos 0" is the number of possible distributions with only the number zero. The row "Pos 1" counts the distributions of zeros and ones. The row "Pos 2" the possible distributions of {0,1,2} and so forth.
From top to down: If a number in the column length = k has reached the value of the sum of the column length = k-1, this number will be k!-(k-1)!+1 times repeated. Before this limit is reached each number is the sum of the neighbor one step above and the neighbor one step to the left.

Examples

			For a(0) we get two possible sequences:
  {0}, {1}.
For a(1) we get three possible sequences:
  {0, 0}, {0, 1}, {1, 1}.
For a(2) = 7 we get:
  {0, 0, 0}, {0, 0, 1}, {0, 0, 2}, {0, 1, 1},
  {0, 1, 2}, {1, 1, 1}, {1, 1, 2}.

Crossrefs

Cf. A000108 (if we change the definition into 0 <= b(k) <= k).
Cf. A005269, A005270, A008934, A016121, A128094, A242105.

Programs

  • PARI
    a(n) = binomial((n-1)! + n-1, n-1) + binomial((n-1)! + n-2, n-1) + sum(r = 1, n-2, sum(k = 0, r-1 ,binomial((n-1)! - r! - k+n - 2, n-1)*binomial(r-1,k)*a(r)*(-1)^(k+1)))

Formula

a(n) = binomial((n-1)! + n-1, n-1) + binomial((n-1)! + n-2, n-1) + Sum_{r = 1..n-2} Sum_{k = 0..r-1} binomial((n-1)! - r! - k+n - 2, n-1)*binomial(r-1,k)*a(r)*(-1)^(k+1).
Showing 1-8 of 8 results.

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

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