The generating function enumerating n by, 4, 0-1 matrices avoiding both verti\
    cal and horizontal patterns 010, 101 (i.e. the Mot Alone restrticion) is



      20      19        18       17        16        15        14         13
- (9 t   + 9 t   - 116 t   + 31 t   + 304 t   - 315 t   - 249 t   + 1570 t

            12         11         10         9        8         7        6
     - 980 t   - 2884 t   + 2696 t   + 1660 t  - 408 t  - 1860 t  - 423 t

            5        4       3       2              /   20    19       18
     + 547 t  + 256 t  + 43 t  - 58 t  - 7 t - 1)  /  (t   + t   - 14 t
                                                  /

          17       16       15       14        13        12        11        10
     + 3 t   + 42 t   - 35 t   - 43 t   + 194 t   - 104 t   - 420 t   + 348 t

            9        8        7       6        5       4    3       2
     + 336 t  - 120 t  - 312 t  - 83 t  + 115 t  + 58 t  - t  - 12 t  - 3 t + 1

    )



                              and in Maple notatin



-(9*t^20+9*t^19-116*t^18+31*t^17+304*t^16-315*t^15-249*t^14+1570*t^13-980*t^12-\
2884*t^11+2696*t^10+1660*t^9-408*t^8-1860*t^7-423*t^6+547*t^5+256*t^4+43*t^3-58
*t^2-7*t-1)/(t^20+t^19-14*t^18+3*t^17+42*t^16-35*t^15-43*t^14+194*t^13-104*t^12
-420*t^11+348*t^10+336*t^9-120*t^8-312*t^7-83*t^6+115*t^5+58*t^4-t^3-12*t^2-3*t
+1)


            For the sake of the OEIS here are the first, 50, terms



[10, 100, 378, 2030, 9484, 46746, 225654, 1098136, 5327258, 25875154, 125619088
, 609970274, 2961628194, 14380186660, 69822244298, 339019686614, 1646096608476,
7992561729954, 38807580840854, 188428757462264, 914908750899146, 
4442305123352810, 21569445781449640, 104729634598331642, 508510810379079306, 
2469055156642623012, 11988404654002419282, 58209248897770412110, 
282632823546329375444, 1372313068094537875962, 6663214601996913714558, 
32352988443054943024312, 157088721243090478852394, 762738390774251269418050, 
3703447632377009671982288, 17981950996116362984037922, 
87310688235434464376906970, 423933770133823040453287780, 
2058394511508829590555026154, 9994457303253258364502982934, 
48527712364201905261712551556, 235624486237603651942047615650, 
1144065850416844470124426730254, 5554968802224301469574565454424, 
26971942552469497515105920991602, 130961254861128559821657940899082, 
635877458267505134661854781442664, 3087479135424484198161476782402586, 
14991139075213629851515220834744986, 72788913192606634996281621483985172]


                          ---------------------------



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