A093678 Sequence contains no 3-term arithmetic progression, starting with 1, 7.

1, 7, 8, 10, 11, 16, 17, 20, 28, 34, 35, 37, 38, 43, 44, 47, 82, 88, 89, 91, 92, 97, 98, 101, 109, 115, 116, 118, 119, 124, 125, 128, 244, 250, 251, 253, 254, 259, 260, 263, 271, 277, 278, 280, 281, 286, 287, 290, 325, 331, 332, 334, 335, 340, 341, 344, 352

Offset: 1

Author

Ralf Stephan, Apr 09 2004

Keywords

• nonn
• look

Comments

a(1)=1, a(2)=7; a(n) is least k such that no three terms of a(1), a(2), ..., a(n-1), k form an arithmetic progression.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries related to non-averaging sequences

Crossrefs

Cf. A004793, A033157, A093679, A093680, A093681, A092482.

Row 3 of array in A093682.

Programs

Maple

N:= 1000: # to get all terms <= N
V:= Vector(N,1):
A[1]:= 1: A[2]:= 7: k:= 8;
for n from 3 while k < N do
  for k from 1 to n-2 do
    p:= 2*A[n-1]-A[k];
    if p <= N then V[p]:= 0 fi
  od:
  for k from A[n-1]+1 to N do
    if V[k] = 1 then A[n]:= k; nmax:= n; break fi;
  od;
od:
seq(A[i],i=1..nmax); # Robert Israel, May 07 2018

Mathematica

a[n_] := Sum[(1/2)(3^IntegerExponent[k, 2]+1), {k, 1, n-1}] + (1/8)( 12(-1)^n - 7Sin[n Pi/2] + 7Sin[3n Pi/2] - Sin[(n+1)Pi/4] + Sin[(5n+1) Pi/4] + Cos[n Pi/2] + Cos[3n Pi/2] + Cos[n Pi/4] + Cos[3n Pi/4] + Cos[5n Pi/4] + Cos[7n Pi/4] + Cos[(3n+1)Pi/4] - Cos[(7n+1)Pi/4] + 38); Array[a, 60] (* Jean-François Alcover, Mar 22 2019 *)

Formula

a(n) = (Sum_{k=1..n-1} (3^A007814(k) + 1)/2) + f(n), with f(n) an 8-periodic function with values {1, 6, 5, 6, 2, 6, 5, 7, ...}, as proved by Lawrence Sze.