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

1,2

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

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.

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 *)

Crossrefs

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

Row 3 of array in A093682.

Sequence in context: A120192 A333490 A256651 * A188052 A266727 A214004

Adjacent sequences: A093675 A093676 A093677 * A093679 A093680 A093681

Keyword

nonn,look

Author

Ralf Stephan, Apr 09 2004

Status

approved