I wonder if building this table can be sped up by noticing a recurring pattern?

    x    0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 ... 254 255
    f(x) 0  0  1  2  3  3  4  5  6  6  7  8  9  9 10 11 12 12 ... 190 191
So something like

    sta table,y
    iny
    sta table,y
    adc $0
    iny    
    sta table,y
    adc $0
    iny    
    sta table,y
    adc $0
    iny    
used as the loop body that should be repeated 64 times, should work. Will it take less than 6000 cycles total?

I didn't check your code worked.. just copied it and ran the inner part of the loop once, but according to https://www.masswerk.at/6502/

It's about 2x faster. Your code uses 44 CPU cycles x 64

Edit: plus a branch instruction, maybe that adds 3 cycles x 64 I guess