Many Elementary Cellular Automata have attributes that generate this sequence. See the Wolfram, Weisstein and Index links below.
Rules 151,159,183,191,215,222,223,247,254 and 255 produce a triangle read by rows giving successive states of cellular automaton initiated with a single ON (black) cell.
Rules 16,24,48,56,80,88,112,120,144,152,176,184,208,216,240 and 248 produce a binary representation of the n-th iteration of the elementary cellular automaton starting with a single ON (black) cell.
Rules 16,24,48,56,80,88,112,120,144,152,176,184,208,216,240 and 248 produce a decimal representation of the n-th iteration of the elementary cellular automaton starting with a single ON (black) cell.
Rules 4,5,12,13,28,29,36,44,60,68,69,70,71,76,77,78,79,92,93,100,102,108,110,124,132,133,140,141,150,151,156,157,159,164,164,172,173,182,183,188,189,191,196,197,199,204,205,206,207,215,220,221,222,223,228,229,230,231,236,237,238,239,247,253,254 and 255 produce a middle column of the elementary cellular automaton starting with a single ON (black) cell.
Rules 2,4,10,12,16,24,34,36,42,44,48,56,66,68,74,76,80,88,98,100,106,108,112,120,130,132,138,140,144,152,162,164,170,172,176,184,194,196,202,204,208,216,226,228,234,236,240 and 248 produce the number of ON (black) cells in the n-th iteration of the elementary cellular automaton starting with a single ON (black) cell.
Rules 0,8,32,40,64,72,96,104,128,136,160,168,192,200,224,232 produce the number of ON (black) cells after n iterations of the elementary cellular automaton starting with a single ON (black) cell.
Rules 2,6,10,14,18,22,26,30,34,38,42,46,50,54,58,62,66,70,74,78,82,86,90,94,98,102,106,110,114,118,122, 126,130,131,134,135,138,139,142,143,146,147,150,151,154,155,158,159,162,163,166,167,170,171,174, 175,178,179,182,183,186,187,190,191,194,195,198,199,202,203,206,207,210,211,214,215,218,219,222, 223,226,227,230,231,234,235,238,239,242,243,246,247,250,251,254 and 255 produce the left diagonal in a binary representation of the elementary cellular automaton starting with a single ON (black) cell.
Rules 16,18,20,22,24,26,28,30,48,50,52,54,56,58,60,62,80,82,84,86,88,90,92,94,112,114,116,118,120, 122,124,126,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,176,177,178,179, 180,181,182,183,184,185,186,187,188,189,190,191,208,209,210,211,212,213,214,215,216,217,218, 219,220,221,222,223,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254 and 255 produce the right diagonal in a binary representation of the elementary cellular automaton starting with a single ON (black) cell.