Many Elementary Cellular Automata (see Wolfram and Weisstein links) have attributes that generate this sequence.
Rules 0,8,32,40,64,72,96,104,128,136,160,168,192,200,224 and 232 produce a triangle read by rows giving successive states of cellular automaton initiated with a single ON (black) cell.
Rules 0,8,32,40,64,72,96,104,128,136,160,168,192,200,224 and 232 produce a binary representation of 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 and 232 produce a decimal representation of the n-th iteration of the elementary cellular automaton starting with a single ON (black) cell.
Rules 0,2,8,10,16,18,24,26,32,34,40,42,48,56,64,66,72,74,80,82,88,90,96,98,104, 106,112,120,128,130,136,138,144,146,152,153,154,160,162,168,170,176,184,192,
194,195,200,202,203,208,210,216,217,218,219,224,226,232,234,240 and 248 produce a middle column 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 and 232 produce the number of ON (black) cells in the n-th iteration of the elementary cellular automaton starting with a single ON (black) cell.