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

1,2

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

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).

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 *)

Prog

(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

Crossrefs

Cf. A004793, A033157.

Sequence in context: A309815 A256800 A172105 * A335099 A147303 A346593

Adjacent sequences: A092479 A092480 A092481 * A092483 A092484 A092485

Keyword

nonn

Author

Ralf Stephan, Apr 04 2004

Extensions

Name clarified by Charles R Greathouse IV, Jan 30 2014

Status

approved