A301844Number of 4Xn 0..1 arrays with every element equal to 0, 1 or 2 horizontally or antidiagonally adjacent elements, with upper left element zero.

8, 128, 773, 5748, 39703, 281758, 1986213, 14047365, 99396932, 703490647, 4979239708, 35242914039, 249447707376, 1765580587312, 12496708381182, 88451201375638, …

The first 2 terms appear at position 147.

…3844609550582231725359408128481117450284102701938521…
position 147
termsstringfirst positionoccurrences
1811100,001,839
2812814799,813
3812877313,085,462100
481287735748not within 1,000,000,0000

Searched the first 1,000,000,000 digits. Strings longer than 12 digits are not indexed.