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A343264 Cardinalities of the sets of fusible numbers obtained at the consecutive steps of their construction as follows. We set S(0) = {0}. S(n+1) is obtained by adding to S(n) the sums (x+y+1)/2 for all x,y from S(n) with the property |x-y| < 1. Then, a(n) is the number of elements in S(n).

%I A343264 #29 Nov 15 2024 19:52:59
%S A343264 1,2,4,9,21,50,119,281,656,1513,3449,7777,17363,38422,84355,183915,
%T A343264 398526,858901
%N A343264 Cardinalities of the sets of fusible numbers obtained at the consecutive steps of their construction as follows. We set S(0) = {0}. S(n+1) is obtained by adding to S(n) the sums (x+y+1)/2 for all x,y from S(n) with the property |x-y| < 1. Then, a(n) is the number of elements in S(n).
%H A343264 Jeff Erickson, Gabriel Nivasch and Junyan Xu, <a href="https://arxiv.org/abs/2003.14342">Fusible numbers and Peano Arithmetic</a>, arXiv:2003.14342 [cs.LO], 2020.
%H A343264 David A. Corneth, <a href="/A343264/a343264.gp.txt">PARI program</a>
%e A343264 a(1) = 2 because S(1) = {0, 1/2};
%e A343264 a(2) = 4 because S(2) = {0, 1/2, 3/4, 1};
%e A343264 a(3) = 9 because S(3) = {0, 1/2, 3/4, 7/8, 1, 9/8, 5/4, 11/8, 3/2}.
%p A343264 s:= proc(n) option remember; `if`(n=0, {0}, (l-> (m-> {seq([2*x, seq(
%p A343264      `if`(abs(x-y)<m, x+y+m, [][]), y=l)][], x=l)})(2^(n-1)))(s(n-1)))
%p A343264     end:
%p A343264 a:= n-> nops(s(n)):
%p A343264 seq(a(n), n=0..10);  # _Alois P. Heinz_, Apr 09 2021
%t A343264 S[n_]:=S[n]=If[n==0,{0},S[n-1]\[Union]Map[(#[[1]]+#[[2]]+1)/2&,Select[Tuples[S[n-1],{2}],Abs[#[[1]]-#[[2]]]<1&]]]; Table[Length[S[n]],{n,0,12}]
%o A343264 (PARI) \\ See Corneth link. _David A. Corneth_, Apr 09 2021
%Y A343264 Cf. A283075.
%K A343264 nonn,more
%O A343264 0,2
%A A343264 _Mamuka Jibladze_, Apr 09 2021
%E A343264 a(13) from _Alois P. Heinz_, Apr 09 2021
%E A343264 a(14)-a(17) from _David A. Corneth_, Apr 10 2021