A003278 -id:A003278 - cp's OEIS frontend

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

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

T. D. Noe, Apr 09 2014

nonn

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?

			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.

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.

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

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

T. D. Noe, Apr 09 2014

nonn

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

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.

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

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

Definition corrected by M. F. Hasler, Jan 12 2016

A055246 At step number k >= 1 the 2^(k-1) open intervals that are erased from [0,1] in the Cantor middle-third set construction are I(k,n) = (a(n)/3^k, (1+a(n))/3^k), n=1..2^(k-1).

Original entry on oeis.org

1, 7, 19, 25, 55, 61, 73, 79, 163, 169, 181, 187, 217, 223, 235, 241, 487, 493, 505, 511, 541, 547, 559, 565, 649, 655, 667, 673, 703, 709, 721, 727, 1459, 1465, 1477, 1483, 1513, 1519, 1531, 1537, 1621, 1627, 1639, 1645, 1675, 1681, 1693, 1699

Offset: 1

Wolfdieter Lang, May 23 2000

Related to A005836. Gives boundaries of open intervals that have to be erased in the Cantor middle-third set construction.
Let g(n) = Sum_{i=0..n} (i*binomial(n+i,i)^3*binomial(n,i)^2) = A112035(n). Let b = {m>0 : g(m) != 0 (mod 3)}. Then b(n) = a(n). - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 08 2004
Conjecture: Similarly to A191107, this increasing sequence is generated by the rules: a(1) = 1, and if x is in the sequence, then 3*x-2 and 3*x+4 are also in the sequence. - L. Edson Jeffery, Nov 17 2015

			k=1: (1/3, 2/3);
k=2: (1/9, 2/9), (7/9, 8/9);
k=3: (1/27, 2/27), (7/27, 8/27), (19/27, 20/27), (25/27, 26/27); ...

Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Index entries for 3-automatic sequences.

Cf. A003278, A005836, A005823, A055247, A112035.

Cf. A191107.

(* (Conjectured) Choose rows large enough to guarantee that all terms < max are generated. *)
rows = 1000; max = 10^4; a[1] = {1}; i = 1; Do[a[n_] = {}; Do[If[1 < 3*a[n - 1][[k]] - 2 < max, AppendTo[a[n], 3*a[n - 1][[k]] - 2], Break]; If[3*a[n - 1][[k]] + 4 < max, AppendTo[a[n], 3*a[n - 1][[k]] + 4], Break], {k, Length[a[n - 1]]}]; If[a[n] == {}, Break, i++], {n, 2, 1000}]; a055246 = Take[Flatten[Table[a[n], {n, i}]], 48] (* L. Edson Jeffery, Nov 17 2015 *)
Join[{1}, 1 + 6 Accumulate[Table[(3^IntegerExponent[n, 2] + 1)/2, {n, 60}]]] (* Vincenzo Librandi, Nov 26 2015 *)

g(n)=sum(i=0,n,i*binomial(n+i,i)^3*binomial(n,i)^2);
for (i=1,2000,if(Mod(g(i),3)<>0,print1(i,",")))

a(n) = fromdigits(binary(n-1),3)*6 + 1; \\ Kevin Ryde, Apr 23 2021

def A055246(n): return int(bin(n-1)[2:],3)*6|1 # Chai Wah Wu, Jun 26 2025

a(n) = 1+6*A005836(n), n >= 1.
a(n) = 1+3*A005823(n), n >= 1.
a(n+1) = A074938(n) + A074939(n); A074938: odd numbers in A005836, A074939: even numbers in A005836. - Philippe Deléham, Jul 10 2005
Conjecture: a(n) = 2*A191107(n) - 1 = 6*A003278(n) - 5 = (a((2*n-1)*2^(k-1))+2)/3^k, k>0. - L. Edson Jeffery, Nov 25 2015

Edited by N. J. A. Sloane, Nov 20 2015: used first comment to give more precise definition, and edited a comment at the suggestion of L. Edson Jeffery.

A309890 Lexicographically earliest sequence of positive integers without triples in weakly increasing arithmetic progression.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 4, 4, 1, 1, 2, 1, 1, 2, 2, 4, 4, 2, 4, 4, 5, 5, 8, 5, 5, 9, 1, 1, 2, 1, 1, 2, 2, 4, 4, 1, 1, 2, 1, 1, 2, 2, 4, 4, 2, 4, 4, 5, 5, 8, 5, 5, 9, 2, 4, 4, 5, 5, 10, 5, 5, 10, 10, 11, 13, 10, 11, 10, 11, 13, 10, 10, 12, 13, 10, 13, 11, 12, 20, 11, 1, 1, 2, 1, 1, 2, 2, 4, 4, 1, 1, 2, 1, 1, 2, 2, 4, 4, 2

Offset: 1

Sébastien Palcoux, Aug 21 2019

Formal definition: lexicographically earliest sequence of positive integers a(n) such that for any i > 0, there is no n > 0 such that 2a(n+i) = a(n) + a(n+2i) AND a(n) <= a(n+i) <= a(n+2i).
Sequence suggested by Richard Stanley as a variant of A229037. They differ from the 55th term. The numbers n for which a(n) = 1 are given by A003278, or equally by A005836 (Richard Stanley).
The sequence defined by c(n) = 1 if a(n) = 1 and otherwise c(n) = 0 is A039966 (although with a different offset). - N. J. A. Sloane, Dec 01 2019
Pleasant to listen to (button above) with Instrument no. 13: Marimba (and for better listening, save and convert to MP3).

Sébastien Palcoux, Table of n, a(n) for n = 1..100000
Sébastien Palcoux, On the first sequence without triple in arithmetic progression (version: 2019-08-21), second part, MathOverflow
Sébastien Palcoux, Table of n, a(n) for n = 1..1000000
Sébastien Palcoux, Density plot of the first 1000000 terms

Cf. A003278, A005836, A039966, A229037.

from itertools import count, islice
def A309890_gen(): # generator of terms
    blist = []
    for n in count(0):
        i, j, b = 1, 1, set()
        while n-(i<<1) >= 0:
            x, y = blist[n-2*i], blist[n-i]
            z = (y<<1)-x
            if x<=y<=z:
                b.add(z)
                while j in b:
                    j += 1
            i += 1
        blist.append(j)
        yield j
A309890_list = list(islice(A309890_gen(),30)) # Chai Wah Wu, Sep 12 2023

# %attach SAGE/ThreeFree.spyx
from sage.all import *
cpdef ThreeFree(int n):
     cdef int i,j,k,s,Li,Lj
     cdef list L,Lb
     cdef set b
     L=[1,1]
     for k in range(2,n):
          b=set()
          for i in range(k):
               if 2*((i+k)/2)==i+k:
                    j=(i+k)/2
                    Li,Lj=L[i],L[j]
                    s=2*Lj-Li
                    if s>0 and Li<=Lj:
                         b.add(s)
          if 1 not in b:
               L.append(1)
          else:
               Lb=list(b)
               Lb.sort()
               for t in Lb:
                    if t+1 not in b:
                         L.append(t+1)
                         break
     return L

A191107 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-2 and 3x+1 are in a.

Original entry on oeis.org

1, 4, 10, 13, 28, 31, 37, 40, 82, 85, 91, 94, 109, 112, 118, 121, 244, 247, 253, 256, 271, 274, 280, 283, 325, 328, 334, 337, 352, 355, 361, 364, 730, 733, 739, 742, 757, 760, 766, 769, 811, 814, 820, 823, 838, 841, 847, 850, 973, 976, 982, 985, 1000, 1003, 1009, 1012, 1054, 1057, 1063, 1066, 1081, 1084, 1090, 1093, 2188

Offset: 1

Clark Kimberling, May 26 2011

For general discussions, see A190803 and A191106.

Numbers whose base-3 representation ends in 1 and contains no 2; primitive members of A005836. - Peter Munn, Aug 14 2023

Robert Israel, Table of n, a(n) for n = 1..10000
Barry Brent, On the Constant Terms of Certain Laurent Series, Preprints (2023) 2023061164.

Cf. A003278, A005836, A055246, A190640, A190803, A191106.

N:= 100000: # to get all terms <= N
with(queue):
Q:= new(1):
A:= {}:
while not empty(Q) do
  s:= dequeue(Q);
  A:= A union {s};
  for t in {3*s-2,3*s+1} minus A do
    if t <= N then enqueue(Q,t) fi
  od
od:
sort(convert(A,list)); # Robert Israel, Nov 29 2015

h = 3; i = -2; j = 3; k = 1; f = 1;  g = 7;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A191107 *)
b = (a + 2)/3; c = (a - 1)/3; r = Range[1, 900];
d = Intersection[b, r] (* A003278 *)
e = Intersection[c, r] (* A005836 *)

Conjecture: a(n) = 3*A003278(n) - 2 = (A055246(n) + 1)/2. - L. Edson Jeffery, Nov 25 2015
Conjecture: a(n) = A190640(n)/2. - Michel Marcus, Aug 24 2016
Conjecture: a(n) = A003278(2n-1). - Arie Bos, Aug 07 2022

A026471 a(n) = least positive integer > a(n-1) and not of the form a(i) + a(j) + a(k) for 1 <= i < j < k <= n.

Original entry on oeis.org

1, 2, 3, 4, 5, 13, 14, 15, 25, 26, 27, 37, 38, 48, 49, 50, 60, 61, 71, 72, 73, 83, 84, 94, 95, 96, 106, 107, 117, 118, 119, 129, 130, 140, 141, 142, 152, 153, 163, 164, 165, 175, 176, 186, 187, 188, 198, 199, 209, 210, 211, 221, 222, 232, 233, 234, 244, 245, 255

Offset: 1

Clark Kimberling

Wieb Bosma, Rene Bruin, Robbert Fokkink, Jonathan Grube, Anniek Reuijl, and Thian Tromp, Using Walnut to solve problems from the OEIS, arXiv:2503.04122 [math.NT], 2025.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A003278, A033627, A075122, A075123.

{1, 5, 13} union {n congruent 2, 3, 4, 14, 15 mod 23}, proved by Matthew Akeran. - Ralf Stephan, Nov 15 2004

G.f.: (9*x^11-7*x^10+9*x^8+7*x^5+x^4+x^3+x^2+x+1)*x/(x^6-x^5-x+1). - Alois P. Heinz, Aug 06 2018

Edited by Floor van Lamoen, Sep 02 2002

A101886 Smallest natural number sequence without any length 4 equidistant arithmetic subsequences.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 14, 15, 16, 18, 19, 20, 22, 24, 27, 28, 29, 31, 32, 35, 36, 37, 39, 41, 42, 43, 47, 48, 50, 51, 53, 55, 58, 60, 61, 63, 65, 66, 68, 70, 71, 72, 77, 78, 80, 82, 85, 86, 87, 89, 90, 91, 94, 95, 96, 98, 99, 100, 102, 103, 104, 107, 109, 110, 111, 114

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 20 2004

nonn

			4 is out because of 1,2,3,4. 13 is out because of 1,5,9,13.

Cf. A101887, A101884, A101888.

A selection of sequences related to "no three-term arithmetic progression": A003002, A003003, A003278, A004793, A005047, A005487, A033157, A065825, A092482, A093678, A093679, A093680, A093681, A093682, A094870, A101884, A101886, A101888, A140577, A185256, A208746, A229037.

A101888 Smallest natural number sequence without any length 5 equidistant arithmetic subsequences.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 22, 23, 24, 25, 27, 28, 29, 30, 32, 33, 34, 35, 37, 38, 39, 40, 43, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 56, 58, 59, 60, 61, 64, 65, 66, 67, 69, 70, 71, 72, 74, 75, 76, 77, 79, 80, 81, 82, 86, 87, 88, 90, 91, 92, 93, 95

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 20 2004

nonn

			5 is out because of 1,2,3,4,5. 21 is out because of 1,6,11,16,21.

Cf. A101889, A101884, A101886.

A selection of sequences related to "no three-term arithmetic progression": A003002, A003003, A003278, A004793, A005047, A005487, A033157, A065825, A092482, A093678, A093679, A093680, A093681, A093682, A094870, A101884, A101886, A101888, A140577, A185256, A208746, A229037.

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

M. F. Hasler, Jan 12 2016

nonn

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.

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

M. F. Hasler, Jan 12 2016

nonn

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.

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions

A055246 At step number k >= 1 the 2^(k-1) open intervals that are erased from [0,1] in the Cantor middle-third set construction are I(k,n) = (a(n)/3^k, (1+a(n))/3^k), n=1..2^(k-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula

Extensions

A309890 Lexicographically earliest sequence of positive integers without triples in weakly increasing arithmetic progression.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

SageMath

A191107 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-2 and 3x+1 are in a.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A026471 a(n) = least positive integer > a(n-1) and not of the form a(i) + a(j) + a(k) for 1 <= i < j < k <= n.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A101886 Smallest natural number sequence without any length 4 equidistant arithmetic subsequences.

Views

Author

Keywords

Examples

Crossrefs

A101888 Smallest natural number sequence without any length 5 equidistant arithmetic subsequences.

Views

Author

Keywords

Examples

Crossrefs

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

Views

Author