A033163 -id:A033163 - cp's OEIS frontend

A185256 Stanley Sequence S(0,3).

0, 3, 4, 7, 9, 12, 13, 16, 27, 30, 31, 34, 36, 39, 40, 43, 81, 84, 85, 88, 90, 93, 94, 97, 108, 111, 112, 115, 117, 120, 121, 124, 243, 246, 247, 250, 252, 255, 256, 259, 270, 273, 274, 277, 279, 282, 283, 286, 324, 327, 328, 331, 333, 336, 337, 340, 351, 354, 355, 358, 360, 363

Offset: 1

N. J. A. Sloane, Mar 19 2011

nonn

Given a finite increasing sequence V = [v_1, ..., v_k] containing no 3-term arithmetic progression, the Stanley Sequence S(V) is obtained by repeatedly appending the smallest term that is greater than the previous term and such that the new sequence also contains no 3-term arithmetic progression.

			After [0, 3, 4, 7, 9] the next term cannot be 10 or we would have the 3-term A.P. 4,7,10; it cannot be 11 because of 7,9,11; but 12 is OK.

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

Alois P. Heinz, Table of n, a(n) for n = 1..2048
P. Erdos et al., Greedy algorithm, arithmetic progressions, subset sums and divisibility, Discrete Math., 200 (1999), 119-135.
J. L. Gerver and L. T. Ramsey, Sets of integers with no long arithmetic progressions generated by the greedy algorithm, Math. Comp., 33 (1979), 1353-1359.
R. A. Moy, On the growth of the counting function of Stanley sequences, arXiv:1101.0022 [math.NT], 2010-2012.
R. A. Moy, On the growth of the counting function of Stanley sequences, Discrete Math., 311 (2011), 560-562.
A. M. Odlyzko and R. P. Stanley, Some curious sequences constructed with the greedy algorithm, 1978.
S. Savchev and F. Chen, A note on maximal progression-free sets, Discrete Math., 306 (2006), 2131-2133.
Index entries for non-averaging sequences

For other examples of Stanley Sequences see A005487, A005836, A187843, A188052, A188053, A188054, A188055, A188056, A188057.

See also A004793, A033160, A033163.

# Stanley Sequences, Discrete Math. vol. 311 (2011), see p. 560
ss:=proc(s1,M) local n,chvec,swi,p,s2,i,j,t1,mmm; t1:=nops(s1); mmm:=1000;
s2:=Array(1..t1+M,s1); chvec:=Array(0..mmm);
for i from 1 to t1 do chvec[s2[i]]:=1; od;
# Get n-th term:
for n from t1+1 to t1+M do # do 1
# Try i as next term:
for i from s2[n-1]+1 to mmm do # do 2
swi:=-1;
# Test against j-th term:
for j from 1 to n-2 do # do 3
p:=s2[n-j];
if 2*p-i < 0 then break; fi;
if chvec[2*p-i] = 1 then swi:=1; break; fi;
od; # od 3
if swi=-1 then s2[n]:=i; chvec[i]:=1; break; fi;
od; # od 2
if swi=1 then ERROR("Error, no solution at n = ",n); fi;
od; # od 1;
[seq(s2[i],i=1..t1+M)];
end;
ss([0,3],80);

ss[s1_, M_] := Module[{n, chvec, swi, p, s2, i, j, t1, mmm}, t1 = Length[s1]; mmm = 1000; s2 = Table[s1, {t1 + M}] // Flatten; chvec = Array[0&, mmm];
For[i = 1, i <= t1, i++, chvec[[s2[[i]] ]] = 1];
(* get n-th term *)
For[n = t1+1, n <= t1 + M, n++,
(* try i as next term *)
For[i = s2[[n-1]] + 1, i <= mmm, i++, swi = -1;
(* test against j-th term *)
For[j = 1, j <= n-2, j++, p = s2[[n - j]]; If[2*p - i < 0, Break[] ];
If[chvec[[2*p - i]] == 1, swi = 1; Break[] ] ];
If[swi == -1, s2[[n]] = i; chvec[[i]] = 1; Break[] ] ];
If[swi == 1, Print["Error, no solution at n = ", n] ] ];
Table[s2[[i]], {i, 1, t1 + M}] ];
ss[{0, 3}, 80] (* Jean-François Alcover, Sep 10 2013, translated from Maple *)

A185256(n,show=1,L=3,v=[0,3], D=v->v[2..-1]-v[1..-2])={while(#v1||next(2), 2); break)); if(type(show)=="t_VEC", v, v[n])} \\ 2nd (optional) arg: zero = silent, nonzero = verbose, vector (e.g. [] or [1]) = get the whole list [a(1..n)] as return value, else just a(n). - M. F. Hasler, Jan 18 2016

A004793 a(1)=1, a(2)=3; a(n) is least k such that no three terms of a(1), a(2), ..., a(n-1), k form an arithmetic progression.

Original entry on oeis.org

1, 3, 4, 6, 10, 12, 13, 15, 28, 30, 31, 33, 37, 39, 40, 42, 82, 84, 85, 87, 91, 93, 94, 96, 109, 111, 112, 114, 118, 120, 121, 123, 244, 246, 247, 249, 253, 255, 256, 258, 271, 273, 274, 276, 280, 282, 283, 285, 325, 327, 328, 330, 334, 336, 337, 339, 352, 354

Offset: 1

N. J. A. Sloane, Clark Kimberling

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000
F. Iacobescu, Smarandache Partition Type and Other Sequences, Bull. Pure Appl. Sci. 16E, 237-240, 1997.
Eric Weisstein's World of Mathematics, Nonarithmetic Progression Sequence
Index entries related to non-averaging sequences

Equals A186776(n)+1, A033160(n)-1, A033163(n)-2.
Cf. A092482, A185256.
Row 1 of array in A093682.

a:= proc(n) local m, r, b; m, r, b:= n-1, 2-irem(n, 2), 1;
      while m>0 do r:= r+b*irem(m, 2, 'm'); b:= b*3 od; r
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 02 2021

Select[Range[1000], MatchQ[IntegerDigits[#-1, 3], {(0|1)..., 0|2}]&] (* Jean-François Alcover, Jan 13 2019, after Tanya Khovanova in A186776 *)

v[1]=1; v[2]=3; for(n=3,1000,f=2; m=v[n-1]+1; while(1, forstep(k=n-1,1,-1,if(v[k]<(m+1)/2,f=1; break); for(l=1,k-1,if(m-v[k]==v[k]-v[l],f=0; break)); if(f<2,break)); if(!f,m=m+1;f=2); if(f==1,break)); v[n]=m) \\ Ralf Stephan

a(n)=if(n<1,1,if(n%2==0,3*a(n/2)-2-3*((n/2)%2),3*a((n-1)/2)-3*(((n-1)/2)%2))) \\ Ralf Stephan

a(n) = (3-n)/2 + 2*floor(n/2) + Sum_{k=1..n-1} 3^A007814(k)/2 = A003278(n) + [n is even], proved by Lawrence Sze, following a conjecture by Ralf Stephan.

a(n) = b(n-1), with b(0)=1, b(2n) = 3b(n) - 2 - 3[n odd], b(2n+1) = 3b(n)-3[n odd].

Rechecked by David W. Wilson, Jun 04 2002

A186776 Stanley Sequence S(0,2).

Original entry on oeis.org

0, 2, 3, 5, 9, 11, 12, 14, 27, 29, 30, 32, 36, 38, 39, 41, 81, 83, 84, 86, 90, 92, 93, 95, 108, 110, 111, 113, 117, 119, 120, 122, 243, 245, 246, 248, 252, 254, 255, 257, 270, 272, 273, 275, 279, 281, 282, 284, 324, 326, 327, 329, 333, 335, 336, 338, 351, 353, 354, 356, 360, 362, 363, 365, 729, 731, 732, 734, 738, 740

Offset: 1

N. J. A. Sloane, Mar 19 2011

nonn

See A185256.
In ternary these numbers have 0's and 1's everywhere, except the last digit is either 0 or 2. [Tanya Khovanova, Nov 16 2013]
a(n) = A005836(n) + A000035(n). [Tanya Khovanova, Nov 16 2013]

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Cf. A185256, A004793, A033160, A033163.

Select[Range[0, 1000], MatchQ[IntegerDigits[#, 3], {(0|1)..., 0|2}]&] (* Jean-François Alcover, Jan 13 2019, after Tanya Khovanova *)

Equals A004793(n)-1, also A033160(n)-2, also A033163(n)-3. Note that A004793 has a recurrence.

A033160 Begins with (2, 4); avoids 3-term arithmetic progressions.

Original entry on oeis.org

2, 4, 5, 7, 11, 13, 14, 16, 29, 31, 32, 34, 38, 40, 41, 43, 83, 85, 86, 88, 92, 94, 95, 97, 110, 112, 113, 115, 119, 121, 122, 124, 245, 247, 248, 250, 254, 256, 257, 259, 272, 274, 275, 277, 281, 283, 284, 286, 326, 328, 329, 331, 335, 337, 338, 340, 353, 355, 356, 358, 362

Offset: 1

N. J. A. Sloane

nonn

F. Iacobescu, Smarandache Partition Type and Other Sequences, Bull. Pure Appl. Sci. 16E, 237-240, 1997.
H. Ibstedt, A Few Smarandache Sequences, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 170-183.
Eric Weisstein's World of Mathematics, Nonarithmetic Progression Sequence

a(n) = A186776(n)+2 = A004793(n)+1 = A033163(n)-1. Cf. A185256.

Select[Range[1000], MatchQ[IntegerDigits[#-2, 3], {(0|1)..., 0|2}]&] (* Jean-François Alcover, Jan 13 2019, after Tanya Khovanova in A186776 *)

A267650 Stanley sequence S_4(0,3): lexicographic first increasing sequence with a(0) = 0, a(1) = 3 and no 4 terms in arithmetic progression.

Original entry on oeis.org

0, 3, 4, 5, 7, 8, 10, 11, 16, 17, 18, 20, 21, 27, 28, 29, 31, 32, 34, 35, 36, 53, 55, 56, 57, 60, 61, 62, 64, 67, 69, 75, 87, 91, 100, 101, 103, 104, 105, 108, 109, 110, 114, 116, 120, 125, 127, 128, 129, 132, 134, 135, 164, 168, 173, 174, 175, 177, 181, 182, 184, 188, 190

Offset: 0

M. F. Hasler, Jan 18 2016

nonn

See A185256 for S(0,3) = S_3(0,3) and A266728 for S_5(0,3).

Index entries related to Stanley sequences

For other examples of Stanley Sequences see A005487, A005836, A187843, A188052, A188053, A188054, A188055, A188056, A188057, A266727, A266728.

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A185256 Stanley Sequence S(0,3).

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A004793 a(1)=1, a(2)=3; a(n) is least k such that no three terms of a(1), a(2), ..., a(n-1), k form an arithmetic progression.

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A186776 Stanley Sequence S(0,2).

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A033160 Begins with (2, 4); avoids 3-term arithmetic progressions.

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A267650 Stanley sequence S_4(0,3): lexicographic first increasing sequence with a(0) = 0, a(1) = 3 and no 4 terms in arithmetic progression.

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