A370708 - OEIS

%I #44 Mar 31 2024 02:13:33

%S 1,2,3,5,6,8,12,13,15,16,21,23,28,32,37,38,40,45,47,61,63,70,73,80,81,

%T 91,96,100,103,105,116,123,128,134,138,150,156,157,175,179,181,190,

%U 207,210,214,217,226,240,243,252,256,265,275,281,283,289,292,293,308,315

%N a(1)=1; thereafter a(n) is the smallest number > a(n-1) such that no two triples of earlier terms in arithmetic progression have the same common difference.

%C A triple consists of three distinct values in a(1), a(2), ..., a(n).

%C By definition, no arithmetic progression of length > 3 can occur in the sequence.

%C What is the density of this sequence?

%H Michael S. Branicky, <a href="/A370708/b370708.txt">Table of n, a(n) for n = 1..10000</a>

%e 4 is not a term in the sequence because it would create the arithmetic progression (2,3,4), which has the same common difference (1) as the previously occurring triple (1,2,3).

%e 9 is not a term because it would create the arithmetic progression (3,6,9), which has the same common difference (3) as the previously occurring (2,5,8).

%o (Python)

%o from itertools import islice

%o def cd(k, alst, dset, diff_dict):

%o     newdset = set()

%o     for a in alst:

%o         if k-a in diff_dict[a]:

%o             if k-a in dset:

%o                 return False

%o             else:

%o                 newdset.add(k-a)

%o     return True, newdset

%o def agen(): # generator of terms

%o     alst, dset, an = [1, 2, 3], {1}, 3

%o     yield from alst

%o     diff_dict = {1: set(), 2: {1}, 3: {1, 2}}

%o     while True:

%o         k = an+1

%o         while not (ans:=cd(k, alst, dset, diff_dict)): k += 1

%o         dset.update(ans[1])

%o         an = k

%o         diff_dict[k] = {an-a for a in alst}

%o         alst.append(an)

%o         yield an

%o print(list(islice(agen(), 60))) # _Michael S. Branicky_, Mar 30 2024

%Y Cf. A003278.

%K nonn

%O 1,2

%A _Neal Gersh Tolunsky_, Mar 25 2024

%E a(15) and beyond from _Michael S. Branicky_, Mar 30 2024