1. Find the Next Term in an Integer Sequence__ __ __ __ __ __ __ __ __
For the last sequence, use OEIS to reveal that the nth term is the largest number that can be created by stacking (in exponential form) a set of numbers that add to n. For example, the fifth term is 3^2, the sixth term is 3^3, the seventh term is 2^3^2, the eighth term is 2^3^3 = 134,217,728. For n >= 7, the nth term is f(n) = 2^f(n – 2).
0 1 2 … __
__ [OEIS]
Online Encyclopedia of Integer Sequences:
Oct. 2018
Predicting the Future

2. Find Missing Term in an Arbitrary Sequence
Z O T T F F __
J F M __ M J J
31 __
A E F H I __ __
__ 18 24
1 3/2 __ 7/8 9/16 __
Questions regarding the 1, 11, 21,1211,… series: Will the sequence always contain only the digits 1, 2, and 3? Is the length of the terms nondecreasing?
In the last example, the first digit indicates round (1) or square (2) lens; the second digit, straight (1) or curved (2) bridge; the third digit, handles to the left (1) or to the right (2). So, the answer is 222.
221
111
212
122
___
Oct. 2018
Predicting the Future

3. A Solution Method for Numerical Series
Polynomial interpolation:
You can pass a line through any two points, a hyperbola through any three points, a third-degree curve through any four points, and so on 4 8 12 20 16 __ __
f(n) = an3 + bn2 + cn + d
n = 1: a + b + c + d = 1
n = 2: 8a + 4b + 2c + d = 4
n = 3: 27a + 9b + 3c + d = 9
n = 4: 64a + 16b + 4c + d = 16
b = 1; a = c = d = 0; f(n) = n2 1 2 3 4 5 6 __
Oct. 2018
Predicting the Future

4. Polynomial Extrapolation Example
This exponential series, when solved via polynomial extrapolation, yields a different answer! __
f(n) = an3 + bn2 + cn + d
n = 2: 8a + 4b + 2c + d = 4
n = 3: 27a + 9b + 3c + d = 8
n = 4: 64a + 16b + 4c + d = 16
a = 1/3; b = –1; c = 8/3; d = 0; f(n) = (1/3)n3 – n2 + (8/3)n
f(5) = (1/3)125–25+(8/3)5 = 30
f(6) = (1/3)216–36+(8/3)6 = 52
f(30) = ,180
= 1,073,741,824
Oct. 2018
Predicting the Future