A093678 - OEIS

%I #22 May 24 2022 02:47:14

%S 1,7,8,10,11,16,17,20,28,34,35,37,38,43,44,47,82,88,89,91,92,97,98,

%T 101,109,115,116,118,119,124,125,128,244,250,251,253,254,259,260,263,

%U 271,277,278,280,281,286,287,290,325,331,332,334,335,340,341,344,352

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

%C 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.

%H Robert Israel, <a href="/A093678/b093678.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/No#non_averaging">Index entries related to non-averaging sequences</a>

%F 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.

%p N:= 1000: # to get all terms <= N

%p V:= Vector(N,1):

%p A[1]:= 1: A[2]:= 7: k:= 8;

%p for n from 3 while k < N do

%p   for k from 1 to n-2 do

%p     p:= 2*A[n-1]-A[k];

%p     if p <= N then V[p]:= 0 fi

%p   od:

%p   for k from A[n-1]+1 to N do

%p     if V[k] = 1 then A[n]:= k; nmax:= n; break fi;

%p   od;

%p od:

%p seq(A[i],i=1..nmax); # _Robert Israel_, May 07 2018

%t 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 *)

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

%Y Row 3 of array in A093682.

%K nonn,look

%O 1,2

%A _Ralf Stephan_, Apr 09 2004