sub popnum1 {
    my( $x, $y, $z ) = @_;

    return $aob[ $x ][ $y ] = 
        $y && substr( $aod[ $y - 1 ], $x, 1 ) ne 'a'
            ? $initial + $z
            : $z + $aob[$x][ $y - 1 ];
}