A057926 a(0) = 1, a(1) = 3, a(n) = smallest number such that sequence b(n) = a(n) OR a(n+1) is strictly monotonically increasing.

1, 3, 4, 8, 5, 10, 16, 11, 20, 32, 21, 34, 24, 35, 28, 64, 29, 66, 32, 67, 36, 72, 37, 74, 48, 75, 52, 128, 53, 130, 56, 131, 60, 192, 61, 194, 256, 195, 260, 200, 261, 202, 272, 203, 276, 224, 277, 226, 280, 227, 284, 512, 285, 514, 288, 515, 292, 520, 293, 522, 304