A033158 -id:A033158 - cp's OEIS frontend

A093682 Array T(m,n) by antidiagonals: nonarithmetic-3-progression sequences with simple closed forms.

1, 2, 1, 4, 3, 1, 5, 4, 4, 1, 10, 6, 5, 7, 1, 11, 10, 8, 8, 10, 1, 13, 12, 10, 10, 11, 19, 1, 14, 13, 13, 11, 13, 20, 28, 1, 28, 15, 14, 16, 14, 22, 29, 55, 1, 29, 28, 17, 17, 20, 23, 31, 56, 82, 1, 31, 30, 28, 20, 22, 28, 32, 58, 83, 163, 1, 32, 31, 31, 28, 23, 29, 37, 59, 85

Offset: 0

Ralf Stephan, Apr 09 2004

The nonarithmetic-3-progression sequences starting with a(1)=1, a(2)=1+3^m or 1+2*3^m, m >= 0, seem to have especially simple 'closed' forms. None of these formulas have been proved, however.

T(m,1)=1, T(m,2) = 1 + (1 + [m even])*3^floor(m/2) = 1 + A038754(m), m >= 0, n > 0; T(m,n) is least k such that no three terms of T(m,1), T(m,2), ..., T(m,n-1), k form an arithmetic progression.

			Array begins:
  1,  2,  4,  5, 10, 11, 13, ...
  1,  3,  4,  6, 10, 12, 13, ...
  1,  4,  5,  8, 10, 13, 14, ...
  1,  7,  8, 10, 11, 16, 17, ...
  1, 10, 11, 13, 14, 20, 22, ...
  ...

Eric Weisstein's World of Mathematics, Nonarithmetic Progression Sequence.
Index entries related to non-averaging sequences

Rows 0-6 are A003278, A004793, A033157, A093678, A093679, A093680, A093681.

Column 2 is 1+A038754. Cf. A092482, A033158.

T(m, n) = (Sum_{k=1..n-1} (3^A007814(k) + 1)/2) + f(n), with f(n) a P-periodic function, where P <= 2^floor((m+3)/2) (conjectured and checked up to m=13, n=1000).

The formula implies that T(m, n) = b(n-1) where b(2n) = 3b(n) + p(n), b(2n+1) = 3b(n) + q(n), with p, q sequences generated by rational o.g.f.s.

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

Original entry on oeis.org

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

N. J. A. Sloane

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

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).

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Thomas Bloom, Problem 271, Erdős Problems.
Paul Erdős, Vsevolod F. Lev, Gérard Rauzy, Csaba Sandor, and Andràs Sarkozy, Greedy algorithm, arithmetic progressions, subset sums and divisibility, Discrete Mathematics 200 (1999), pp. 119-135.
Richard A. Moy and David Rolnick, Novel structures in Stanley sequences, arXiv:1502.06013 [math.CO], 2015.
Richard A. Moy and David 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.
Terence Tao, Erdős problem database, see no. 271.
Index entries related to non-averaging sequences

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

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 *)

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

Name clarified by Charles R Greathouse IV, Jan 30 2014

A236269 First differences of Stanley sequence S(0,4) (A005487).

Original entry on oeis.org

4, 1, 2, 4, 1, 4, 7, 3, 5, 2, 4, 1, 6, 5, 7, 17, 5, 2, 5, 10, 4, 7, 18, 4, 3, 5, 7, 26, 19, 9, 6, 17, 1, 5, 1, 11, 9, 12, 10, 7, 8, 3, 15, 6, 2, 3, 6, 18, 48, 7, 5, 25, 12, 21, 11, 4, 21, 2, 6, 5, 50, 5, 21, 18, 30, 1, 6, 5, 4, 6, 4, 1, 2, 20, 10, 4, 24, 3, 13, 5

Offset: 1

Ralf Stephan, Jan 21 2014

nonn

Also first differences of Stanley sequence S(1,5) (A033158).
While there are conjectures about formulas for S(0,m), m=1,2,3,6,9... (see A093682), m=4 is the first case where the first differences look almost random.
Records are 4, 7, 17, 18, 26, 48, 50, 55, 76, 87, 92, 93, 165, 175,...
Positions of records are 1, 7, 16, 23, 28, 49, 61, 81, 83, 101, 147, 165, 185, 250, 400,...
Positions where a(n)=1: 2, 5, 12, 33, 35, 66, 72, 94, 125, 160, 189, 288, 307, 327,...

Ralf Stephan, Table of n, a(n) for n = 1..4650

NAP(sv,N)=local(v,vv,m,k,l,sl,vvl);sl=length(sv);vvl=min(N*N,10^5);v=vector(N);vv=vector(vvl);for(k=1,sl,v[k]=sv[k];for(l=1,k-1,vv[2*v[k]-v[l]]=1));m=v[sl]+1;for(k=sl+1,N,while(m<=vvl&&vv[m],m=m+1);if(m>vvl,return(v));for(l=1,k-1,sl=2*m-v[l];if(sl<=vvl,vv[sl]=1));vv[m]=1;v[k]=m);v
S04(n)=N=1000;NAP([0,4],N)[n]
a(n)=S04(n+1)-S04(n)

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A093682 Array T(m,n) by antidiagonals: nonarithmetic-3-progression sequences with simple closed forms.

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

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A236269 First differences of Stanley sequence S(0,4) (A005487).

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