A004793 a(1)=1, a(2)=3; a(n) is least k such that no three terms of a(1), a(2), ..., a(n-1), k form an arithmetic progression.

1, 3, 4, 6, 10, 12, 13, 15, 28, 30, 31, 33, 37, 39, 40, 42, 82, 84, 85, 87, 91, 93, 94, 96, 109, 111, 112, 114, 118, 120, 121, 123, 244, 246, 247, 249, 253, 255, 256, 258, 271, 273, 274, 276, 280, 282, 283, 285, 325, 327, 328, 330, 334, 336, 337, 339, 352, 354

Offset: 1

Author

N. J. A. Sloane, Clark Kimberling

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• F. Iacobescu, Smarandache Partition Type and Other Sequences, Bull. Pure Appl. Sci. 16E, 237-240, 1997.
• Eric Weisstein's World of Mathematics, Nonarithmetic Progression Sequence
• Index entries related to non-averaging sequences

Crossrefs

Equals A186776(n)+1, A033160(n)-1, A033163(n)-2.

Cf. A092482, A185256.

Row 1 of array in A093682.

Programs

Maple

a:= proc(n) local m, r, b; m, r, b:= n-1, 2-irem(n, 2), 1;
      while m>0 do r:= r+b*irem(m, 2, 'm'); b:= b*3 od; r
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 02 2021

Mathematica

Select[Range[1000], MatchQ[IntegerDigits[#-1, 3], {(0|1)..., 0|2}]&] (* Jean-François Alcover, Jan 13 2019, after Tanya Khovanova in A186776 *)

PARI

v[1]=1; v[2]=3; for(n=3,1000,f=2; m=v[n-1]+1; while(1, forstep(k=n-1,1,-1,if(v[k]<(m+1)/2,f=1; break); for(l=1,k-1,if(m-v[k]==v[k]-v[l],f=0; break)); if(f<2,break)); if(!f,m=m+1;f=2); if(f==1,break)); v[n]=m) \\ Ralf Stephan

PARI

a(n)=if(n<1,1,if(n%2==0,3*a(n/2)-2-3*((n/2)%2),3*a((n-1)/2)-3*(((n-1)/2)%2))) \\ Ralf Stephan

Formula

a(n) = (3-n)/2 + 2*floor(n/2) + Sum_{k=1..n-1} 3^A007814(k)/2 = A003278(n) + [n is even], proved by Lawrence Sze, following a conjecture by Ralf Stephan.

a(n) = b(n-1), with b(0)=1, b(2n) = 3b(n) - 2 - 3[n odd], b(2n+1) = 3b(n)-3[n odd].

Extensions

Rechecked by David W. Wilson, Jun 04 2002