1. Numbers 1, 2, 3, 4, etc., alternated with their square -1

1, 1*1-1=0, ... , 4, 4*4-1=15

2. Alternate between dividing the previous number by two and multiplying the previous by three.

16, 16/2=8, 8*3=24, ..., 18*3=54

3. 1, 2, 3, 4, etc., after each two numbers insert their sum: of each set is one greater than the first number of the previous set.

1, 2, 1+2=3, 3, 4, 3+4=7, ..., 5+6=11

4. Start with 0. Add 1 to previous number, then 2 to previous, then 3, etc.

0, 0+1=1, 1+2=3, 3+3=6, 6+410, 10+515, 15+621, 21+728

5. Alternate between dividing previous number by 2, dividing it by 4, and multiplying it by 6.

64, 64/2=32, 32/4=8, 8*6=48, 48/2=24, 24/4=6, 6*6=36, 36/2=18

6. Each number is the sum of the previous two; if sum is > 100, then subtract 100.

35, 18, 35+18=53, 18+53=71, 71+53-100=24, 71+24=95, 24+95-100=19, 95+19-100=14

7. Change previous number by 1², 2², 3², 4², ... The "change" alternates between adding and subtracting.

1, 1+1²=2, 2-2²=-2, -2+3²=7, 7-4²=-9, -9+5²=16, 16-6²=-20 -20+7²=29

8. Take prime numbers in descending order, starting at 23. The sequence is these prime numbers squared.

23²=529, 19²=371, 17²=289, 13²=169, 11²=121, 7²=49, 5²=25

9. Take the sequence 1, 2, 3, 4, 5, etc. and alternate between squaring and cubing the number; take the number modulo 100. "x modulo y" means to take the remainder when x is divided by y. For the puzzles, we usually use modulo 100 -- this means to drop all but the tens and ones digits: 625 modulo 100 = 25.

1²=1, 2³=8, 3²=9, 4³=64, 5²=25, 6³ modulo 100=16,
7² modulo 100=49, 8³ modulo 100=12, 9²=81, 10³ modulo 100=0

10. Each number is calculated from the previous two as the absolute difference between them (that is, the larger number minus the smaller).

63, 74, 74-63=11, 74-11=63, 63-11=52, 63-52=11, 52-11=41, 41-11=30