A092482 Sequence contains no 3-term arithmetic progression, other than its initial terms 1, 2, 3.

1, 2, 3, 6, 7, 14, 15, 17, 18, 36, 37, 39, 40, 45, 46, 48, 49, 98, 99, 101, 102, 107, 108, 110, 111, 125, 126, 128, 129, 134, 135, 137, 138, 276, 277, 279, 280, 285, 286, 288, 289, 303, 304, 306, 307, 312, 313, 315, 316, 357, 358, 360, 361, 366, 367, 369, 370

Offset: 1

Author

Ralf Stephan, Apr 04 2004

Keywords

• nonn

Comments

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

Links

• David A. Corneth, Table of n, a(n) for n = 1..8193 (first 512 terms by Jean-François Alcover)
• Eric Weisstein's World of Mathematics, Nonarithmetic Progression Sequence.
• Index entries related to non-averaging sequences

Crossrefs

Cf. A004793, A033157.

Programs

Mathematica

a[n_] := a[n] = If[n < 4, n, For[k = a[n - 1] + 1, True, k++, sp = SequencePosition[Append[Array[a, n - 1], k], {x_, _, y_, _, z_} /; y - x == z - y, 2]; If[sp == {{1, 3}}, Return[k]]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 512}]
(* Comparing with data from conjectured formula: *)
b[n_] := If[n < 4, n, 1 + 2^(Length[id = IntegerDigits[n - 2, 2]] - 1) + FromDigits[id, 3]];
Table[b[n], {n, 1, 512}] (* Jean-François Alcover, Jan 15 2019 *)
(* Second [much faster] program: *)
upto[m_] := Module[{n, v, i, j}, n = Max[m, 3]; v = Table[1, {n}]; For[i = 3, i <= n-1, i++, If[v[[i]] == 1, For[j = Max[1, 2i-n], j <= Min[2n-i, i-1], j++, If[v[[j]] == 1, v[[2i-j]] = 0]]]]; Position[v, 1] // Flatten]; upto[12000] (* Jean-François Alcover, Jan 16 2019, after David A. Corneth *)

PARI

upto(n) = n=max(n,3); v=vector(n, i, 1); for(i=3, n-1, if(v[i]==1, for(j = max(1, 2*i-n), min(2*n-i,i-1), c=2*i - j; if(v[j]==1, v[2*i-j]=0; )))); select(x -> x==1, v, 1) \\ David A. Corneth, Jan 15 2019

Formula

For n > 2, a(n+2) = 1 + 2^floor(log_2(n)) + Sum_{k=1..n} (3^A007814(n) + 1)/2 = 1 + A053644(n) + A005836(n) (conjectured and checked up to n=512).

Extensions

Name clarified by Charles R Greathouse IV, Jan 30 2014