The Life of a Mathematician

Okay, so I'm going to try and start this up again. I have officially relocated to Washington state, and am employed in writing a solutions manual for Mathematics: A Human Endeavor by Harold R. Jacobs, which is a pre-algebra text. The job provides room and board, and I live in an apartment which is attached to a house; one of the guys I'm working for and his wife live downstairs. They also own a vacation home rental business. On Tuesdays (or sometimes Thursdays) and Fridays I do classroom observations, where the curriculum I'm writing solutions for is being field-tested in classrooms (some at a private school and some with homeschoolers), in preparation for filming video lectures. The rest of the time I work from my apartment.

Today I got to give the last half of the lecture to the third homeschool class, which consists of two girls taking prealgebra. I was trying to help them review for a test, and we were going over conic sections. Not having seen the lectures on this material, I was mostly guessing from what was in the review section (no pun intended). In trying to explain the basic concept of conic sections, I used a styrofoam coffee cup as a prop. It's not quite a cone, but its convex hull is a truncated cone and imagination can supply the rest.

Chapter 2 of this Jacobs text is about number sequences. Earlier I was revisiting a number sequence that I've played with in the past. It's a Fibonacci variant which seems to me to be pretty obvious, but I've never seen it mentioned anywhere:

f(0) = f(1) = f(2) = 1
f(n) = f(n-1) + f(n-3), n > 2

1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 278, 407, 596, 874, 1281...

Compare this to the original Fibonacci sequence, also carried far enough to hit four digits:

f(0) = f(1) = 1
f(n) = f(n-1) + f(n-2), n > 1

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597....

One interesting thing about the new sequence is that any two consecutive numbers after the first three are generated by adding two completely different pairs of earlier terms. The sequence can actually be generated by a story problem similar to the original one with the rabbits. Fibonacci's rabbits took one month to mature, and reproduced a new pair every month thereafter. The new rabbits take two months to mature, but still reproduce at the same rate after doing so. Using * for a pair of baby rabbits, r for a pair of juvenile rabbits, and R for a pair of adult rabbits, the first few iterations look like this:

*
r
R
R *
R r *
R R r *
R R R r * *
R R R R r r * * *
R R R R R R r r r * * * *
R R R R R R R R R r r r r * * * * * *

Again, compare this with the original problem, where the intermediate stage does not exist:

*
R
R *
R R *
R R R * *
R R R R R * * *
R R R R R R R R * * * * *
R R R R R R R R R R R R R * * * * * * * *
R R R R R R R R R R R R R R R R R R R R R * * * * * * * * * * * * *

It is of course possible to make other, similar modified problems of the form:

f(0) = f(1) = ... = f(k) = 1
f(n) = f(n-1) + f(n - k - 1), n > k

For the original Fibonacci sequence, k = 1, and for the modification shown above, k = 2. Any such sequence would model a rabbit population where the time required grow to maturity is k times the length of a reproduction cycle. It's late now so the rest of this will have to wait.