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).

1, 2, 4, 9, 21, 50, 119, 281, 656, 1513, 3449, 7777, 17363, 38422, 84355, 183915, 398526, 858901

Offset: 0

Author

Mamuka Jibladze, Apr 09 2021

Keywords

• nonn
• more

Examples

			a(1) = 2 because S(1) = {0, 1/2};
a(2) = 4 because S(2) = {0, 1/2, 3/4, 1};
a(3) = 9 because S(3) = {0, 1/2, 3/4, 7/8, 1, 9/8, 5/4, 11/8, 3/2}.
		

Links

• Jeff Erickson, Gabriel Nivasch and Junyan Xu, Fusible numbers and Peano Arithmetic, arXiv:2003.14342 [cs.LO], 2020.
• David A. Corneth, PARI program

Crossrefs

Cf. A283075.

Programs

Maple

s:= proc(n) option remember; `if`(n=0, {0}, (l-> (m-> {seq([2*x, seq(
     `if`(abs(x-y) nops(s(n)):
seq(a(n), n=0..10);  # Alois P. Heinz, Apr 09 2021

Mathematica

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}]

PARI

\\ See Corneth link. David A. Corneth, Apr 09 2021

Extensions

a(13) from Alois P. Heinz, Apr 09 2021

a(14)-a(17) from David A. Corneth, Apr 10 2021