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 1-10 of 11 results. Next

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

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

A267301 Earliest positive increasing sequence having no 5-term subsequence with constant second differences.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 17, 20, 21, 23, 25, 31, 32, 37, 46, 47, 53, 56, 61, 63, 64, 67, 70, 72, 76, 87, 90, 93, 104, 112, 116, 120, 135, 138, 146, 153, 164, 177, 179, 180, 197, 201, 221, 224, 276, 277, 279, 282, 283, 285, 287, 295, 305, 317, 320, 327, 340, 354, 361, 364, 377, 380, 385, 391, 403, 415, 424, 430, 443
Offset: 1

Views

Author

M. F. Hasler, Jan 12 2016

Keywords

Crossrefs

Cf. A267300 (nonnegative variant: 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.
Cf. A240556 and A240557 for sequences avoiding 5-term subsequences with constant third differences.

Programs

  • PARI
    A267301(n, show=0, L=5, 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

A267302 Earliest nonnegative increasing sequence having no 6-term subsequence with constant second differences.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 17, 19, 20, 21, 24, 25, 27, 34, 35, 38, 40, 42, 45, 46, 48, 53, 54, 55, 63, 67, 73, 74, 80, 82, 83, 84, 86, 87, 89, 90, 92, 94, 102, 107, 108, 110, 117, 128, 133, 136, 139, 143, 144, 149, 150, 151, 152
Offset: 1

Views

Author

M. F. Hasler, Jan 12 2016

Keywords

Crossrefs

Cf. A267303 (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. A267300 and A267301 for sequences avoiding 5-term subsequences with constant second differences.
Cf. A240556 and A240557 for sequences avoiding 5-term subsequences with constant third differences.

Programs

  • PARI
    A267302(n, show=0, L=6, 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

A267303 Earliest positive increasing sequence having no 6-term subsequence with constant second differences.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 18, 20, 21, 22, 25, 26, 28, 35, 36, 39, 41, 43, 46, 47, 49, 54, 55, 56, 64, 68, 74, 75, 81, 83, 84, 85, 87, 88, 90, 91, 93, 95, 103, 108, 109, 111, 118, 129, 134, 137, 140, 144, 145, 150, 151, 152, 153
Offset: 1

Views

Author

M. F. Hasler, Jan 12 2016

Keywords

Crossrefs

Cf. A267302 (nonnegative variant: 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.
Cf. A267300 and A267301 for sequences avoiding 5-term subsequences with constant second differences.
Cf. A240556 and A240557 for sequences avoiding 5-term subsequences with constant third differences.

Programs

  • PARI
    A267303(n, show=0, L=6, 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

A267304 Earliest nonnegative increasing sequence having no 7-term subsequence with constant second differences.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 24, 26, 28, 29, 31, 33, 35, 38, 40, 41, 43, 49, 50, 52, 53, 58, 59, 62, 63, 64, 69, 70, 72, 73, 77, 81, 82
Offset: 1

Views

Author

M. F. Hasler, Jan 12 2016

Keywords

Crossrefs

Cf. A267305 (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. A267300 and A267301 for sequences avoiding 5-term subsequences with constant second differences.
Cf. A267302 and A267303 for sequences avoiding 6-term subsequences with constant second differences.
Cf. A240556 and A240557 for sequences avoiding 5-term subsequences with constant third differences.

Programs

  • PARI
    A267304(n, show=0, L=7, 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

A267305 Earliest positive increasing sequence having no 7-term subsequence with constant second differences.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 25, 27, 29, 30, 32, 34, 36, 39, 41, 42, 44, 50, 51, 53, 54, 59, 60, 63, 64, 65, 70, 71, 73, 74, 78, 82, 83
Offset: 1

Views

Author

M. F. Hasler, Jan 12 2016

Keywords

Crossrefs

Cf. A267304 (nonnegative variant: 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.
Cf. A267300 and A267301 for sequences avoiding 5-term subsequences with constant second differences.
Cf. A267302 and A267303 for sequences avoiding 6-term subsequences with constant second differences.
Cf. A240556 and A240557 for sequences avoiding 5-term subsequences with constant third differences.

Programs

  • PARI
    A267305(n, show=0, L=7, 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

A267306 Earliest nonnegative increasing sequence having no 6-term subsequence with constant third differences.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 28, 29, 31, 32, 33, 34, 35, 37, 38, 40, 47, 79, 93, 94, 96, 97, 98, 99, 100, 102, 103, 105, 110, 116, 122, 128, 130, 140, 148, 266, 281, 296, 303, 304, 306, 308, 311, 313, 318, 324, 326, 327, 328, 330, 331, 332, 337
Offset: 1

Views

Author

M. F. Hasler, Jan 12 2016

Keywords

Crossrefs

Cf. A267307 (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. A267300 and A267301 for sequences avoiding 5-term subsequences with constant second differences.
Cf. A267302 and A267303 for sequences avoiding 6-term subsequences with constant second differences.
Cf. A267304 and A267305 for sequences avoiding 7-term subsequences with constant second differences.
Cf. A240556 and A240557 for sequences avoiding 5-term subsequences with constant third differences.

Programs

  • PARI
    A267306(n, show=0, L=6, 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

Extensions

More terms from Jinyuan Wang, Jan 01 2021
Showing 1-10 of 11 results. Next

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