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A163687
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Number of nX4 binary arrays with all 1s connected, a path of 1s from upper left corner to lower right corner, and no 1 having more than two 1s adjacent
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0
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1, 8, 39, 132, 389, 1095, 3107, 8790, 24449, 66724, 179212, 475169, 1247181, 3249223, 8420475, 21735923, 55926076, 143501956, 367381497, 938864371, 2396062211, 6108616160, 15560958273
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