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 11-20 of 27 results. Next

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).

A005839 Lexicographically earliest increasing nonnegative sequence that contains no 4-term arithmetic progression.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 9, 14, 15, 16, 18, 25, 26, 28, 29, 30, 33, 36, 48, 49, 50, 52, 53, 55, 56, 57, 62, 64, 65, 66, 79, 86, 87, 88, 90, 93, 98, 101, 104, 105, 108, 109, 110, 121, 125, 135, 144, 148, 150, 151, 159, 162, 166, 168, 169, 170, 173, 175, 176, 182
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) = A005837(n) - 1. - Alois P. Heinz, Jan 31 2014

References

  • R. K. Guy, Unsolved Problems in Number Theory, E10.
  • 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

  • Mathematica
    t = {0, 1, 2}; Do[s = Table[Append[i, n], {i, Subsets[t, {3}]}];
If[! MemberQ[Table[Differences[i, 2], {i, s}], {0, 0}], AppendTo[t, n]], {n, 3, 200}]; t (* T. D. Noe, Apr 17 2014 *)

Extensions

More terms from Jeffrey Shallit, Aug 15 1995.
Edited (with new offset, etc.) by N. J. A. Sloane, Jan 04 2016

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

A240556 Earliest nonnegative increasing sequence with no 5-term subsequence of constant third differences.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 11, 15, 16, 27, 47, 48, 64, 95, 175, 196, 211, 212, 214, 247, 249, 252, 398, 839, 1002, 1014, 1016, 1035, 1036, 1037, 1051, 1054, 1072, 1121, 1143, 1146, 1172, 1258, 4271, 4282, 4284, 4336, 4571, 4578, 4582, 4598, 4613, 4622, 4628, 4646
Offset: 1

Views

Author

T. D. Noe, Apr 09 2014

Keywords

Comments

For the positive sequence, see A240557, which is this sequence plus 1. Is there a simple way of determining this sequence, as in the case of the no 3-term arithmetic progression?

Examples

			After (0, 1, 2, 3, 5, 7), the number 10 is excluded since else the subsequence (0, 2, 3, 5, 10) would have successive 1st, 2nd and 3rd differences (2, 1, 2, 5), (-1, 1, 3) and (2, 2), which is constant and thus excluded.

Crossrefs

Cf. A240557 (starting with 1).
No 3-term AP: A005836 (>=0), A003278 (>0);
no 4-term AP: A240075 (>=0), A240555 (>0);
no 5-term AP: A020654 (>=0), A020655 (>0);
no 6-term AP: A020656 (>=0), A005838 (>0);
no 7-term AP: A020657 (>=0), A020658 (>0);
no 8-term AP: A020659 (>=0), A020660 (>0);
no 9-term AP: A020661 (>=0), A020662 (>0);
no 10-term AP: A020663 (>=0), A020664 (>0).
Cf. A240075 and A240555 for sequences avoiding 4-term subsequences with constant second differences.

Programs

  • Mathematica
    t = {0, 1, 2, 3}; Do[s = Table[Append[i, n], {i, Subsets[t, {4}]}]; If[! MemberQ[Flatten[Table[Differences[i, 4], {i, s}]], 0], AppendTo[t, n]], {n, 4, 5000}]; t
  • PARI
    A240556(n,show=0,L=5,o=3,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

A240557 Earliest positive increasing sequence with no 5-term subsequence of constant third differences.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 16, 17, 28, 48, 49, 65, 96, 176, 197, 212, 213, 215, 248, 250, 253, 399, 840, 1003, 1015, 1017, 1036, 1037, 1038, 1052, 1055, 1073, 1122, 1144, 1147, 1173, 1259, 4272, 4283, 4285, 4337, 4572, 4579, 4583, 4599, 4614, 4623, 4629, 4647
Offset: 1

Views

Author

T. D. Noe, Apr 09 2014

Keywords

Comments

For the nonnegative sequence, see A240556, which is this sequence minus 1. Is there a simple way of determining this sequence, as in the case of the no 3-term arithmetic progression?
See crossreferences for sequences avoiding arithmetic progressions. - M. F. Hasler, Jan 12 2016

Crossrefs

Cf. A240556 (starting with 0).
No 3-term AP: A005836 (>=0), A003278 (>0);
no 4-term AP: A240075 (>=0), A240555 (>0);
no 5-term AP: A020654 (>=0), A020655 (>0);
no 6-term AP: A020656 (>=0), A005838 (>0);
no 7-term AP: A020657 (>=0), A020658 (>0);
no 8-term AP: A020659 (>=0), A020660 (>0);
no 9-term AP: A020661 (>=0), A020662 (>0);
no 10-term AP: A020663 (>=0), A020664 (>0).
Cf. A240075 and A240555 for sequences avoiding 4-term subsequences with constant second differences.

Programs

  • Mathematica
    t = {1, 2, 3, 4}; Do[s = Table[Append[i, n], {i, Subsets[t, {4}]}]; If[! MemberQ[Flatten[Table[Differences[i, 4], {i, s}]], 0], AppendTo[t, n]], {n, 5, 5000}]; t
  • PARI
    A240557(n,show=0,L=5,o=3,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 M. F. Hasler, Jan 12 2016

A267300 Earliest positive increasing sequence having no 5-term subsequence with constant second differences.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 8, 11, 13, 16, 19, 20, 22, 24, 30, 31, 36, 45, 46, 52, 55, 60, 62, 63, 66, 69, 71, 75, 86, 89, 92, 103, 111, 115, 119, 134, 137, 145, 152, 163, 176, 178, 179, 196, 200, 220, 223, 275, 276, 278, 281, 282, 284, 286, 294, 304, 316, 319, 326, 339, 353, 360, 363, 376, 379, 384, 390, 402, 414, 423, 429, 442
Offset: 1

Views

Author

M. F. Hasler, Jan 12 2016

Keywords

Crossrefs

Cf. A267301 (positive variant: starting with 1).
No 3-term AP: A005836 (>=0), A003278 (>0);
no 4-term AP: A240075 (>=0), A240555 (>0);
no 5-term AP: A020654 (>=0), A020655 (>0);
no 6-term AP: A020656 (>=0), A005838 (>0);
no 7-term AP: A020657 (>=0), A020658 (>0);
no 8-term AP: A020659 (>=0), A020660 (>0);
no 9-term AP: A020661 (>=0), A020662 (>0);
no 10-term AP: A020663 (>=0), A020664 (>0).
Cf. A240075 and A240555 for sequences avoiding 4-term subsequences with constant second differences.
Cf. A240556 and A240557 for sequences avoiding 5-term subsequences with constant third differences.

Programs

  • PARI
    A267300(n, show=0, L=5, 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
Previous Showing 11-20 of 27 results. Next

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

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