If you go east from Amarillo, north of the I-40, toward the Oklahoma line you'll go through--not by, as happens if you go on the 40--towns with names like Panhandle (appropriately enough), White Deer, Pampa, Mobeetie, Brisco, Allison. Here are some pictures of White Deer (pop. ~1,000)
and Pampa (pop. 18,000). If there are more pictures of Pampa, that's because there's more there there. And that's probably because it's the county seat.
The Texas panhandle has its contradictions. Along with the cowboy/John Deere/religious iconography, there are signs for ending world hunger and a highway memorializing Woody Guthrie.
The panhandle seems to go on forever, every way you look. But eventually it ends, and shortly after crossing the OK line, you spill on to the Washita Battlefield National Historic site, which is worth a stop and look.