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.

Previous Showing 11-19 of 19 results.

A020660 Lexicographically earliest increasing sequence of positive numbers that contains no arithmetic progression of length 8.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 59, 60, 61, 62, 63, 64, 65, 67, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 81, 84, 85, 87, 88, 89, 91, 92, 93, 95, 96, 97
Offset: 1

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
3-term AP: A005836 (>=0), A003278 (>0);
4-term AP: A005839 (>=0), A005837 (>0);
5-term AP: A020654 (>=0), A020655 (>0);
6-term AP: A020656 (>=0), A005838 (>0);
7-term AP: A020657 (>=0), A020658 (>0);
8-term AP: A020659 (>=0), A020660 (>0);
9-term AP: A020661 (>=0), A020662 (>0);
10-term AP: A020663 (>=0), A020664 (>0).

Programs

  • Maple
    Noap:= proc(N,m)
# N terms of earliest increasing seq with no m-term arithmetic progression
local A,forbid,n,c,ds,j;
A:= Vector(N):
A[1..m-1]:= <($1..m-1)>:
forbid:= {m}:
for n from m to N do
  c:= min({$A[n-1]+1..max(max(forbid)+1, A[n-1]+1)} minus forbid);
  A[n]:= c;
  ds:= convert(map(t -> c-t, A[m-2..n-1]),set);
  for j from m-2 to 2 by -1 do
    ds:= ds intersect convert(map(t -> (c-t)/j, A[m-j-1..n-j]),set);
    if ds = {} then break fi;
  od;
  forbid:= select(`>`,forbid,c) union map(`+`,ds,c);
od:
convert(A,list)
end proc:
Noap(100, 8); # Robert Israel, Jan 04 2016

A020661 Lexicographically earliest increasing sequence of nonnegative numbers that contains no arithmetic progression of length 9.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 30, 31, 32, 33, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 52, 54, 55, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 70, 77, 78, 79, 80, 81, 82, 83, 84, 86, 87, 90, 91, 93, 94, 95, 96, 97
Offset: 1

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
3-term AP: A005836 (>=0), A003278 (>0);
4-term AP: A005839 (>=0), A005837 (>0);
5-term AP: A020654 (>=0), A020655 (>0);
6-term AP: A020656 (>=0), A005838 (>0);
7-term AP: A020657 (>=0), A020658 (>0);
8-term AP: A020659 (>=0), A020660 (>0);
9-term AP: A020661 (>=0), A020662 (>0);
10-term AP: A020663 (>=0), A020664 (>0).

A020662 Lexicographically earliest increasing sequence of positive numbers that contains no arithmetic progression of length 9.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 53, 55, 56, 57, 58, 59, 60, 64, 65, 66, 67, 68, 69, 70, 71, 78, 79, 80, 81, 82, 83, 84, 85, 87, 88, 91, 92, 94, 95, 96, 97
Offset: 1

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
3-term AP: A005836 (>=0), A003278 (>0);
4-term AP: A005839 (>=0), A005837 (>0);
5-term AP: A020654 (>=0), A020655 (>0);
6-term AP: A020656 (>=0), A005838 (>0);
7-term AP: A020657 (>=0), A020658 (>0);
8-term AP: A020659 (>=0), A020660 (>0);
9-term AP: A020661 (>=0), A020662 (>0);
10-term AP: A020663 (>=0), A020664 (>0).

Programs

  • Maple
    Noap:= proc(N,m)
# N terms of earliest increasing seq with no m-term arithmetic progression
local A,forbid,n,c,ds,j;
A:= Vector(N):
A[1..m-1]:= <($1..m-1)>:
forbid:= {m}:
for n from m to N do
  c:= min({$A[n-1]+1..max(max(forbid)+1, A[n-1]+1)} minus forbid);
  A[n]:= c;
  ds:= convert(map(t -> c-t, A[m-2..n-1]),set);
  for j from m-2 to 2 by -1 do
    ds:= ds intersect convert(map(t -> (c-t)/j, A[m-j-1..n-j]),set);
    if ds = {} then break fi;
  od;
  forbid:= select(`>`,forbid,c) union map(`+`,ds,c);
od:
convert(A,list)
end proc:
Noap(100,9); # Robert Israel, Jan 04 2016

A020663 Lexicographically earliest increasing sequence of nonnegative numbers that contains no arithmetic progression of length 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 48, 49, 50, 51, 52, 53, 54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 80, 81, 82, 83, 84, 87, 88, 95, 96
Offset: 1

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
3-term AP: A005836 (>=0), A003278 (>0);
4-term AP: A005839 (>=0), A005837 (>0);
5-term AP: A020654 (>=0), A020655 (>0);
6-term AP: A020656 (>=0), A005838 (>0);
7-term AP: A020657 (>=0), A020658 (>0);
8-term AP: A020659 (>=0), A020660 (>0);
9-term AP: A020661 (>=0), A020662 (>0);
10-term AP: A020663 (>=0), A020664 (>0).

A005837 Lexicographically earliest increasing sequence of positive numbers that contains no 4-term arithmetic progression.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 10, 15, 16, 17, 19, 26, 27, 29, 30, 31, 34, 37, 49, 50, 51, 53, 54, 56, 57, 58, 63, 65, 66, 67, 80, 87, 88, 89, 91, 94, 99, 102, 105, 106, 109, 110, 111, 122, 126, 136, 145, 149, 151, 152, 160, 163, 167, 169, 170, 171, 174, 176, 177, 183, 187, 188, 194, 196
Offset: 1

Views

Author

N. J. A. Sloane, Jeffrey Shallit

Keywords

Comments

a(n) = A005839(n) + 1. - Alois P. Heinz, Jan 31 2014

References

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

Links

Crossrefs

Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
3-term AP: A005836 (>=0), A003278 (>0);
4-term AP: A005839 (>=0), A005837 (>0);
5-term AP: A020654 (>=0), A020655 (>0);
6-term AP: A020656 (>=0), A005838 (>0);
7-term AP: A020657 (>=0), A020658 (>0);
8-term AP: A020659 (>=0), A020660 (>0);
9-term AP: A020661 (>=0), A020662 (>0);
10-term AP: A020663 (>=0), A020664 (>0).

Programs

  • Maple
    Noap:= proc(N,m)
# N terms of earliest increasing seq with no m-term arithmetic progression
local A,forbid,n,c,ds,j;
A:= Vector(N):
A[1..m-1]:= <($1..m-1)>:
forbid:= {m}:
for n from m to N do
  c:= min({$A[n-1]+1..max(max(forbid)+1, A[n-1]+1)} minus forbid);
  A[n]:= c;
  ds:= convert(map(t -> c-t, A[m-2..n-1]),set);
  for j from m-2 to 2 by -1 do
    ds:= ds intersect convert(map(t -> (c-t)/j, A[m-j-1..n-j]),set);
    if ds = {} then break fi;
  od;
  forbid:= select(`>`,forbid,c) union map(`+`,ds,c);
od:
convert(A,list)
end proc:
Noap(100,4); # Robert Israel, Jan 04 2016
  • Mathematica
    t = {1, 2, 3}; Do[s = Table[Append[i, n], {i, Subsets[t, {3}]}]; If[! MemberQ[Table[Differences[i, 2], {i, s}], {0, 0}], AppendTo[t, n]], {n, 4, 200}]; t (* T. D. Noe, Apr 17 2014 *)

Extensions

Edited by M. F. Hasler, Jan 03 2016. Further edited (with new offset) by N. J. A. Sloane, Jan 04 2016

A240075 Lexicographically earliest nonnegative increasing sequence such that no four terms have constant second differences.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 15, 16, 17, 20, 44, 51, 52, 53, 56, 58, 64, 78, 166, 167, 192, 195, 196, 200, 202, 203, 206, 217, 226, 248, 249, 276, 312, 649, 657, 678, 681, 682, 715, 726, 740, 743, 747, 750, 771, 790, 830, 833, 836, 838, 842, 854, 875, 908, 911, 971
Offset: 1

Views

Author

T. D. Noe, Apr 09 2014

Keywords

Links

Crossrefs

For the positive sequence, see A240555, which is this sequence plus 1.
Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
3-term AP: A005836 (>=0), A003278 (>0);
4-term AP: A005839 (>=0), A005837 (>0);
5-term AP: A020654 (>=0), A020655 (>0);
6-term AP: A020656 (>=0), A005838 (>0);
7-term AP: A020657 (>=0), A020658 (>0);
8-term AP: A020659 (>=0), A020660 (>0);
9-term AP: A020661 (>=0), A020662 (>0);
10-term AP: A020663 (>=0), A020664 (>0).
For the analog sequence which avoids 5-term subsequences of constant third differences, see A240556 (>=0) and A240557 (>0).

Programs

  • Mathematica
    t = {0, 1, 2}; Do[s = Table[Append[i, n], {i, Subsets[t, {3}]}]; If[! MemberQ[Flatten[Table[Differences[i, 3], {i, s}]], 0], AppendTo[t, n]], {n, 3, 1000}]; t
  • PARI
    A240075(n, show=0, L=4, o=2, v=[0], D=v->v[2..-1]-v[1..-2])={ my(d, m); while( #v1, ); #Set(d)>1||next(2), 2); break)); v[#v]} \\ M. F. Hasler, Jan 12 2016

Extensions

Definition corrected by N. J. A. Sloane and M. F. Hasler, Jan 04 2016.

A240555 Lexicographically earliest positive increasing sequence such that no four terms have constant second differences.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 16, 17, 18, 21, 45, 52, 53, 54, 57, 59, 65, 79, 167, 168, 193, 196, 197, 201, 203, 204, 207, 218, 227, 249, 250, 277, 313, 650, 658, 679, 682, 683, 716, 727, 741, 744, 748, 751, 772, 791, 831, 834, 837, 839, 843, 855, 876, 909, 912, 972
Offset: 1

Views

Author

T. D. Noe, Apr 09 2014

Keywords

Comments

If "positive" is changed to "nonnegative" we get A240075, which is this sequence minus 1.
See A005837 for the earliest sequence containing no 4-term arithmetic progression.

Examples

			After 1,2,3 the number 4 is excluded since (1,2,3,4) has zero second and third differences.
After 1,2,3,5 the number 8 is excluded since (2,3,5,8) has second differences 1,1.

Links

Crossrefs

Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
3-term AP: A005836 (>=0), A003278 (>0);
4-term AP: A005839 (>=0), A005837 (>0);
5-term AP: A020654 (>=0), A020655 (>0);
6-term AP: A020656 (>=0), A005838 (>0);
7-term AP: A020657 (>=0), A020658 (>0);
8-term AP: A020659 (>=0), A020660 (>0);
9-term AP: A020661 (>=0), A020662 (>0);
10-term AP: A020663 (>=0), A020664 (>0).
Cf. A240075 (nonnegative version, a(n)-1).
Cf. A240556 and A240557 for sequences avoiding 5-term subsequences with constant third differences.

Programs

  • Mathematica
    t = {1, 2, 3}; Do[s = Table[Append[i, n], {i, Subsets[t, {3}]}]; If[! MemberQ[Flatten[Table[Differences[i, 3], {i, s}]], 0], AppendTo[t, n]], {n, 4, 1000}]; t
  • PARI
    A240555(n, show=0, L=4, o=2, v=[1], D=v->v[2..-1]-v[1..-2])={ my(d, m); while( #v1, ); #Set(d)>1||next(2), 2); break)); v[#v]} \\ M. F. Hasler, Jan 12 2016

Extensions

Definition corrected by N. J. A. Sloane, Jan 04 2016 and M. F. Hasler at the suggestion of Lewis Chen

A248625 Lexicographically earliest sequence of nonnegative integers such that no triple (a(n),a(n+d),a(n+2d)) is in arithmetic progression, for any d>0, n>=0.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 3, 3, 0, 0, 1, 0, 0, 1, 1, 3, 3, 1, 3, 3, 4, 4, 7, 4, 4, 8, 0, 0, 1, 0, 0, 1, 1, 3, 3, 0, 0, 1, 0, 0, 1, 1, 3, 3, 1, 3, 3, 4, 4, 7, 4, 4, 8, 8, 3, 3, 4, 4, 9, 4, 4, 9, 1, 9, 12, 10, 9, 7, 10, 12, 9, 11, 9, 9, 11, 9, 10, 13, 19, 12, 0, 0, 1, 0, 0, 1, 1, 3, 3
Offset: 0

Views

Author

M. F. Hasler, Oct 10 2014

Keywords

Comments

The sequence is constructed in the greedy way, appending at each step the least nonnegative integer such that no subsequence of equidistant terms contains an AP.
Every nonnegative integer seems to appear in this sequence - see A248627 for the corresponding indices.
Sequence A229037 is the analog for positive integers (and indices).

Examples

			Start with a(0)=a(1)=0, smallest possible choice and trivially satisfying the constraint since no 3-term subsequence is possible.
Then one must take a(2)=1 since otherwise [0,0,0] would be an AP.
Then one can take again a(3)=a(4)=0, and a(5)=1.
Now appending 0 would yield the AP (0,0,0) by extracting terms with indices 0,3,6; therefore a(6)=1.
Now a(7) cannot be 0 not 1 nor 2 since else a(3)=0, a(5)=1, a(7)=2 would be an AP, therefore a(7)=3 is the least possible choice.

Crossrefs

Cf. A005836, A005839, A023717, ..., A020664.
Cf. A248627, A229037, A241752.

Programs

  • PARI
    [DD(v)=vecextract(v,"^1")-vecextract(v,"^-1"), hasAP(a,m=3)=u=vector(m,i,1);v=vector(m,i,i-1);for(i=1,#a-m+1,for(s=1,(#a-i)\(m-1),#Set(DD(t=vecextract(a,(i)*u+s*v)))==1&&return
([i,s,t])))]; a=[]; for(n=1,90,a=concat(a,0);while(hasAP(a),a[#a]++);print1(a[#a]","));a

Formula

a(n) = A229037(n+1)+1.

A071711 Let s(k) denote the k-th term of an integer sequence such that s(0)=0 and s(i) for all i>0 is the least natural number such that no four elements of {s(0),..,s(i)} are in arithmetic progression. Then it appears that there are many set of 3 consecutive integers in s(k). Sequence gives the smallest element in those triples.

Original entry on oeis.org

0, 7, 14, 28, 48, 55, 64, 86, 108, 168, 286, 371, 471, 633, 760, 982, 1032, 1136, 1261, 1600, 1739, 1788, 1822, 1848, 3832, 4225, 5504, 7729, 8062, 9229, 10110, 21977, 27953, 39335, 50820, 50852, 86357, 95586, 106331, 160418, 295806, 314853, 368358, 459825
Offset: 1

Views

Author

Benoit Cloitre, Jun 03 2002

Keywords

Comments

Presumably there are infinitely many such triples.

Links

Crossrefs

Cf. A005839.

Extensions

More terms from Rémy Sigrist, Mar 14 2023
Previous Showing 11-19 of 19 results.

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

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