Measures of Intelligence - Natural and Artificial Reasoning - page 143

Table 11.8 First stage of
sequence differentiation to
that of a constant number

Differential (depth)
(exponent)

Sequence

0 611182738516683102123

1 579111315171921

2 22222222

Example:

(6 11 18 27 38 51 66 83 102 123)

This sequence was originally generated by the function:

x 2 2x 1

F(x) = + + 3

2 x 1

= +

d/dx F(x) 2

d 2 /dx F(x) = 2

Step 1 . Differentiate the sequence until a constant difference is achieved (Table 11.8 ):
Step 2 . The final differential (depth) gives the first exponent.
In this case it is x 2 .
Step 3 . The coefficient for the first exponent is obtained by taking the ratio of the
final number, in this case 2, with the factorial of the exponent (also 2).

2 / (1 ∗ 2)

= 1

so the first term will be x 2 .
Step 4 . For each element in the sequence its values will be:

149162536496481100

Step 4 . These values are subtracted from each element in the sequence. This is
because:

(x 2 2x 1 x 2 2x 1

+ + 3) − = + 3

(6 − 1) (11 − 4) (18 − 9) (27 − 16) (38 − 25) (51 − 36) (66 − 49) (83 − 64)

(102 − 81) (123 − 100)

= 57911131517192123

So this is the sequence for (2 x 1

3)
Step 6. The procedure is repeated for this next set of numbers (Table 11.9 ).

+