This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A050295 #24 Feb 16 2025 08:32:40
%S A050295 1,2,3,5,8,16,24,48,76,132,198,396,588,1176,1764,2940,4680,9360,13680,
%T A050295 27360,43776,72960,109440,218880,330240,660480,990720,1693440,2709504,
%U A050295 5419008,8128512,16257024,25823232,43038720,64558080,129116160,194365440,388730880
%N A050295 Number of strongly triple-free subsets of {1, 2, ..., n}.
%C A050295 A set S is strongly triple-free if x in S implies 2x not in S and 3x not in S.
%C A050295 Conjecture: for k=1,2,3,..., a(6k+1)=2a(6k) and a(6k+5)=2a(6k+4) (these relations hold through a(35)). - _John W. Layman_, Jun 22 2002
%C A050295 From Pradhan Prashanth Kumar (pradhan.ptr(AT)gmail.com), Feb 03 2008:
%C A050295 The conjecture is true. Proof:
%C A050295 Let b(6k+1) = Number of strongly triple-free subsets of {1,2,...,6k+1} which do not contain 6k+1 and c(6k+1) = Number of strongly triple-free subsets of {1,2,...,6k+1} which contain 6k+1. Now a(6k+1) = b(6k+1) + c(6k+1) and b(6k+1) = a(6k).
%C A050295 1) c(6k+1)<=a(6k) : Take any strongly triple-free subset of {1,2,..,6k+1}, which contains 6k+1 and delete 6k+1. The new set is a subset of {1,2,...,6k} and is trongly triple-free. Hence c(6k+1)<=a(6k).
%C A050295 2) c(6k+1)>=a(6k) : Take any strongly triple-free subset of {1,2,...,6k}. Add 6k+1 to it. Since 6k+1 is not divisible by 2 or 3, this new set is still strongly triple-free. Hence c(6k+1)>=a(6k).
%C A050295 This shows that c(6k+1) = a(6k) and therefore a(6k+1) = b(6k+1)+c(6k+1) = 2a(6k). QED
%C A050295 Another proof for the conjecture: a(6k+r) = 2a(6k+r-1) when r={1,5} (with a(0)=1) would be: Any positive integer of form (6k+1) or (6k+5) is neither divisible by 2 nor by 3. Hence adding the number (6k+1) or (6k+5) to the each strongly triple-free subset of {1, ..., 6k} or {1, ..., 6k+4} does not violate the property and hence we would have 2a(6k) or 2a(6k+4) such subsets for a(6k+1) or a(6k+5). - _Ramasamy Chandramouli_, Aug 30 2008
%C A050295 A068060 is the weakly triple-free analog of this sequence. - _Steven Finch_, Mar 02 2009
%H A050295 Alois P. Heinz, <a href="/A050295/b050295.txt">Table of n, a(n) for n = 0..75</a>
%H A050295 Steven R. Finch, <a href="/FinchTriple.html">Triple-Free Sets of Integers</a> [From Steven Finch, Apr 20 2019]
%H A050295 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Triple-FreeSet.html">Triple-Free Set.</a>
%Y A050295 Cf. A050291-A050296, A068060.
%K A050295 nonn
%O A050295 0,2
%A A050295 _Eric W. Weisstein_
%E A050295 More terms from _John W. Layman_, Jun 22 2002
%E A050295 a(0)=1 prepended by _Alois P. Heinz_, Jan 17 2019