Tonight, we discussed number sequences. Much of the discussion was taken from "From To Seek Whence Cometh a Sequence" in Fluid Concepts and Creative Analogies by Douglas Hofstadter

The first few terms would give me a hint, and I would make a guess; then I would calculate a few more terms, often having my guess confirmed but sometimes getting thrown for a loop. The more terms, a new guess, and either a confirmation or a return to the drawing-board.

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, ...
(triangle numbers or sum of the first n numbers)
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121,...
(squares or sum of first n odd numbers)
1, 1, 3, 4, 6, 9, 10, 16, 15, 25, ...
(triangles between squares)
1, 2, 2, 3, 3, 3, 4, 4, 4, 4,...
(n, repeated n times)
2, 3, 5, 7, 11, 13,...
(primes)
2, 3, 3, 5, 5, 5, 7, 7, 7, 7,...
(primes repeated n times)
1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4,...
(n, repeated prime-times)
2, 1, 2, 2, 2, ...
(2s with one oddball)
2, 3, 5, 7, 9, 11, 13, 17,...
(primes with one oddball)
2, 3, 5, 7, 9, 11, 13, 17,...
(odds with one oddball)
1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0,...
(zeros at prime locations)
2, 1, 2, 4, 2, 9, 2, 1...
(2s interleaved with squares)
2, 1, 3, 4, 5, 9, 7, 16,...
(primes interleaved with squares)
1, 0, -6, 0, 120, 0, -5040, 0, 362880, 0, ...
(zeros interleaved with the reciprocals of the Taylor series coefficients for sin x)
2, 5, 11, 17, 23,...
(every second prime)
3, 5, 11, 17, 31, 41,...
(every "primeth" prime)
3, 4, 6, 8, 12, 14, 18,...
(prime+1)
1, 4, 27, 256,...
(n to the nth)
1, 1, 2, 3, 5, 8, 13, 21,...
(Fibonacci)
1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8,...
(n copies of a, the nth element in the sequence)
2, 1, 1, 3, 4, 2, 5, 9, 2, 7, 16, 3, 11,...
(3 sequences interleaved)
1, 2, 3, 5, 7, 8, 11, 13, 17, 19,...
(the union of Fibonacci and primes)
2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10,...
(denominators in the continued-fraction expansion of Euler’s constant e)
14, 18, 23, 28, 34, 42, 50, 59, ...
11, 21, 1211, 111221, ...
(one one; two ones; one two, one one; one one, one two, two ones)
0, 1, 2, 720!,...
(n n-factorials)
7, 3, 10, 4, 8, ...
(building numbers in the infinite corridor)
7, 3, 11, 10, 4, 8, ...
(or arguably an alternative interpretation of the infinite corridor numbering)
49, 53, 58, 63, 69, 75, ...
(n+1 = n+int(n/10))

Finding the right derived operation (first differences of terms, counting repeated terms, interwoven sequences, smoothness, rate of growth, periodic behavior, etc.) help to unlock the secret. But people also notice facts about numbers, such as primes and squares—All of which leads to strategies for controlling a search

Breadth-first leads to combinatorial explosions; depth-first leads to sink holes; or the importance of "sniffing before you inhale deeply."

Heuristics: What should be "high on the shelf" vs. ready at hand?

Number Savvy
1. recognizing salient properties, e.g., 3 cubed is 27
2. making plausible guesses, e.g., 3125 might be a power of 5
3. knowing relationships, e.g., all powers of 4 are powers of 2
4. being able to do complicated calculations, e.g., calculating 5 to the 5th

Pattern Sensitivity
1. noticing sameness
2. noticing simple relationships
3. noticing analogies
4. imposing consistencies
5. building abstractions
6. shifting boundaries
7. driving towards beauty

When you are dealing with a world that is much bigger than you can fully explore, you have to make guesses, and guesses are risks, and risks sometimes don’t pan out.

Juxtapose experimentation with theorization!!

Enhance the likelihood of similar perceptions [and] weaken the likelihood of dissimilar perceptual structures.

"God is subtle but not perverse."--approximately Einstein

Themes:
1. the inseparability of perception and high-level cognition [seeing and thinking are inseparable--Lettvin]
2. easily reconfigurable, multi-level cognitive representations
3. subcognitive pressures [some things are more important than others]
4. commingling of many pressures
5. simultaneous feeling-out of many potential pathways
6. the centrality of the making of analogies and variations on a theme
7. deeper and shallower aspects
8. inner structure of concepts and conceptual neighborhoods

"It is my firm belief that pattern perception, extrapolation, and generalization are the true crux of creativity."--Hofstadter

## Puzzles for next week

Sharpen syntax flute fair the ??