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

0,2

Links

Table of n, a(n) for n=0..17.

Jeff Erickson, Gabriel Nivasch and Junyan Xu, Fusible numbers and Peano Arithmetic, arXiv:2003.14342 [cs.LO], 2020.

David A. Corneth, PARI program

Example

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}.

Maple

s:= proc(n) option remember; `if`(n=0, {0}, (l-> (m-> {seq([2*x, seq(
     `if`(abs(x-y)<m, x+y+m, [][]), y=l)][], x=l)})(2^(n-1)))(s(n-1)))
    end:
a:= n-> 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}]

Prog

(PARI) See Corneth link \\ David A. Corneth, Apr 09 2021

Crossrefs

Cf. A283075.

Sequence in context: A322325 A355046 A275864 * A018905 A024537 A171842

Adjacent sequences: A343261 A343262 A343263 * A343265 A343266 A343267

Keyword

nonn,more

Author

Mamuka Jibladze, Apr 09 2021

Extensions

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

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

Status

approved