A005487 Starts 0, 4 and contains no 3-term arithmetic progression.

0, 4, 5, 7, 11, 12, 16, 23, 26, 31, 33, 37, 38, 44, 49, 56, 73, 78, 80, 85, 95, 99, 106, 124, 128, 131, 136, 143, 169, 188, 197, 203, 220, 221, 226, 227, 238, 247, 259, 269, 276, 284, 287, 302, 308, 310, 313, 319, 337, 385, 392, 397, 422, 434, 455, 466, 470

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

This is what would now be called the Stanley Sequence S(0,4). See A185256.

References

• R. K. Guy, Unsolved Problems in Number Theory, E10.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..10000
• P. Erdős, V. Lev, G. Rauzy, C. Sandor, and A. Sarkozy, Greedy algorithm, arithmetic progressions, subset sums and divisibility, Discrete Mathematics 200 (1999), pp. 119-135.
• 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.
• Index entries related to non-averaging sequences

Crossrefs

Equals A033158(n+1)-1. Cf. A185256.

Programs

Mathematica

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, 4}, 80] (* Jean-François Alcover, Sep 10 2013, translated from Maple program given in A185256 *)

Python

A005487_list = [0,4]
for i in range(101-2):
    n, flag = A005487_list[-1]+1, False
    while True:
        for j in range(i+1,0,-1):
            m = 2*A005487_list[j]-n
            if m in A005487_list:
                break
            if m < A005487_list[0]:
                flag = True
                break
        else:
            A005487_list.append(n)
            break
        if flag:
            A005487_list.append(n)
            break
        n += 1 # Chai Wah Wu, Jan 05 2016

Extensions

Name clarified by Charles R Greathouse IV, Jan 30 2014