A266727 Stanley sequence S(0,1,7).

0, 1, 7, 8, 10, 11, 17, 18, 30, 31, 37, 38, 40, 41, 47, 48, 90, 91, 97, 98, 100, 101, 107, 108, 120, 121, 127, 128, 130, 131, 137, 138, 270, 271, 277, 278, 280, 281, 287, 288, 300, 301, 307, 308, 310, 311, 317, 318, 360, 361, 367, 368, 370, 371, 377, 378, 390

Offset

0,3

Comments

Lexicographic first increasing sequence starting (0, 1, 7, ...) and such that no 3 terms are in arithmetic progression. - M. F. Hasler, Jan 18 2016

Links

Chai Wah Wu, Table of n, a(n) for n = 0..8191

R. A. Moy and D. Rolnick, Novel structures in Stanley sequences, arXiv:1502.06013 [math.CO], 2015.

R. A. Moy and D. Rolnick, Novel structures in Stanley sequences, Discrete Math., 339 (2016), 689-698.

A. M. Odlyzko and R. P. Stanley, Some curious sequences constructed with the greedy algorithm, 1978.

Example

a(3) = 8 because no 3 terms among (0,1,7,8) are in AP. a(4) is not 9 because (7,8,9) would be an AP, which is excluded. a(4) = 10 works, a(5) = 11 also.
For a(6), 12 is forbidden because of (8,10,12), 13 because of (7,10,13), 14 because of (8,11,14), 15 because of (7,11,15), 16 because of (0,8,16), thus a(6) = 17 is the smallest possible choice. - M. F. Hasler, Jan 18 2016

Mathematica

s = {0, 1, 7};
next[] := For[k = Last[s]+1; AppendTo[s, 0]; , True, k++, s[[-1]] = k; sp = SequencePosition[s, {___, a_, ___, b_, ___, c_, ___} /; b-a == c-b, 1]; If[sp == {}, Print[k]; Break[]]];
Do[next[], {60}];
s (* Jean-François Alcover, Oct 04 2018 *)

Prog

(Python)
A266727_list = [0, 1, 7]
for i in range(1000):
    n, flag = A266727_list[-1]+1, False
    while True:
        for j in range(i+2, 0, -1):
            m = 2*A266727_list[j]-n
            if m in A266727_list:
                break
            if m < A266727_list[0]:
                flag = True
                break
        else:
            A266727_list.append(n)
            break
        if flag:
            A266727_list.append(n)
            break
        n += 1 # Chai Wah Wu, Jan 05 2016
(PARI) A266727(n, show=1, L=3, v=[0, 1, 7], D=v->v[2..-1]-v[1..-2])={while(#v<=n, show&&print1(v[#v]", "); v=concat(v, v[#v]); while(v[#v]++, forvec(i=vector(L, j, [if(j<L, j, #v), #v]), #Set(D(vecextract(v, i)))>1||next(2), 2); break)); if(type(show)=="t_VEC", v, v[n+1])} \\ 2nd (optional) arg: zero = silent, nonzero = verbose, vector (e.g. [] or [1]) = get the whole list [a(0..n)] as return value, else just a(n). - M. F. Hasler, Jan 18 2016

Crossrefs

Cf. A005836, A188053, A266728.

Sequence in context: A256651 A093678 A188052 * A214004 A037263 A125134

Adjacent sequences: A266724 A266725 A266726 * A266728 A266729 A266730

Keyword

nonn

Author

N. J. A. Sloane, Jan 04 2016

Status

approved