

A263402


Define Z(1) = {1}, and Z(n+1) = Z(n) (+) { x+y, with x and y in Z(n) } for any n>0 (where (+) stands for the symmetric difference of two sets). Then a(n) gives the number of elements in Z(n).


1



1, 2, 3, 7, 10, 22, 42, 87, 170, 342, 686, 1365, 2727, 5468, 10919, 21857, 43680, 87389, 174756, 349539, 699039, 1398115, 2796191, 5592422, 11184795, 22369639, 44739229, 89478503, 178956950, 357913967, 715827858, 1431655793, 2863311503, 5726623097, 11453246088
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OFFSET

1,2


COMMENTS

a(n) can also be interpreted as the number of ON cells at the nth stage of the following automaton:
 At first stage, we have only one ON cell at position 1,
 An ON cell appears at position x+y if the cells at positions x and y are ON,
 An ON cell dies at position x+y if the cells at positions x and y are ON.
a(n) <= 2^(n1) for any n>0.


LINKS

Paul Tek, Table of n, a(n) for n = 1..250
Paul Tek, PERL program for this sequence
Paul Tek, Graphic representation of the elements in the range [1..300] of the first 300 sets Z(n)


FORMULA

a(n) = A000120(z(n)) for any n>0
where z(n) is a binary encoding of Z(n), defined as follows:
 z(1) = 2^1,
 z(n+1) = z(n) XOR A067398(z(n)) for any n>0 (where XOR stands for the binary XOR operator).


EXAMPLE

Z(1) = {1};
Z(2) = {1} (+) {2} = {1,2};
Z(3) = {1,2} (+) {2,3,4} = {1,3,4};
Z(4) = {1,3,4} (+) {2,4,5,6,7,8} = {1,2,3,5,6,7,8};
Hence: a(1) = 1, a(2) = 2, a(3) = 3 and a(4) = 7.


PROG

(Perl) See Links section.
(PARI) lista(nn) = {zprec = Set([1]); print1(#zprec, ", "); for (n=2, nn, zs = setbinop((x, y)>x+y, zprec); zn = setminus(setunion(zprec, zs), setintersect(zprec, zs)); print1(#zn, ", "); zprec = zn; ); } \\ Michel Marcus, Oct 20 2015


CROSSREFS

Cf. A067398.
Sequence in context: A291241 A318406 A079380 * A062113 A130968 A007748
Adjacent sequences: A263399 A263400 A263401 * A263403 A263404 A263405


KEYWORD

nonn


AUTHOR

Paul Tek, Oct 17 2015


STATUS

approved



