sub _norm {
	my($o, @v) = @_;
	my $a = 1;
	while ($a) {
		no warnings qw"uninitialized";
		$a = 0;
		for my $i (keys @v) {
			if (1 < $v[$i]) {
				$v[$i] -= 2;
				$v[$i + 1]++;
				if (2 <= $i) {
					$v[$i - 2]++;
				} elsif (1 <= $i) {
					$v[0]++;
				}
				$a++;
			}
			if (0 < $v[$i] && 0 < $v[$i + 1]) {
				$v[$i]--;
				$v[$i + 1]--;
				$v[$i + 2]++;
				$a++;
			}
		}
	}
	$o->_make(@v);
}

sub add { # return the sum of two Zeckendorf objects
	my($o, $t) = @_;
	my @v = @$o;
	my @w = @$t;
	for my $i (keys @w) {
		$v[$i] += $w[$i];
	}
	$o->_norm(@v);
}

sub twice {
	my($o) = @_;
	$o->add($o);
}

our @poweroftwo = (Zeckendorf->one);

sub fromnat { # create a Zeckendorf object from a perl natural number
	my($o, $g) = @_;
	my $t = 1 + 0 * $g;
	my $i = 0;
	my $z = Zeckendorf->zero;
	while ($t <= $g) {
		my $y = $poweroftwo[$i] //= $poweroftwo[$i - 1]->twice;
		if (0 < ($t & $g)) {
			$z = $z->add($y);
		}
		$t <<= 1;
		$i++;
	}
	$z;
}

}