Sequence solutions
By popular demand, here are the solutions for the open sequences. Some of these are trivial, and some of them are not too difficult.
Fix A_n to be the nth term of the sequence in question. All sequences start with n = 0.
Define Phi (n) to be Euler's totient function, or how many numbers are relatively prime to n; Phi (m) = #n such that n does not divide m.
Sequence #2: (difficult). A_n = 3^n modulo Phi (n).
For those of you who are unfamiliar with modular arithmetic, here is a link.
Wikipedia entry on modular arithmetic.
Difficult. Very difficult. I put this one up because the first two were so easy, and there were some complaints.
Sequence #3. (moderate). Define A_0 = Sin (1/2).
Then for n > 0, A_n = Sin (A_{n-1}), that is, iterate the Sin (x) function.
Sequence #6. (easy). A slight change of pace. #6 is a single substition cryptosystem. Let A = 0, B = 1, ... , Z = 25. Then the complete sequence should spell "Angry Grasshopper".
Sequence #8. (easy/moderate). Recursive, although the first two terms are omitted. A_{-2}=1,A_{-1} = 1, and
A_{n}= ( A_{n-1}+A_{n-2} ) \ A_{n-3}.
I felt that I had to omit A_{-2} and A_{-1}, otherwise the sequence was too easy.
Sequence #10. (moderate). Let p = 314159265, Pi_b be the binary representation of p, read backwards. Then A_n is the n+1th digit of Pi_b.
I tried my best to make sure that these sequences weren't in Sloans, and weren't too much of a Rorschach test. I hope that those of you who did work on the sequences had a fun time. =D
Fix A_n to be the nth term of the sequence in question. All sequences start with n = 0.
Define Phi (n) to be Euler's totient function, or how many numbers are relatively prime to n; Phi (m) = #n such that n does not divide m.
Sequence #2: (difficult). A_n = 3^n modulo Phi (n).
For those of you who are unfamiliar with modular arithmetic, here is a link.
Wikipedia entry on modular arithmetic.
Difficult. Very difficult. I put this one up because the first two were so easy, and there were some complaints.
Sequence #3. (moderate). Define A_0 = Sin (1/2).
Then for n > 0, A_n = Sin (A_{n-1}), that is, iterate the Sin (x) function.
Sequence #6. (easy). A slight change of pace. #6 is a single substition cryptosystem. Let A = 0, B = 1, ... , Z = 25. Then the complete sequence should spell "Angry Grasshopper".
Sequence #8. (easy/moderate). Recursive, although the first two terms are omitted. A_{-2}=1,A_{-1} = 1, and
A_{n}= ( A_{n-1}+A_{n-2} ) \ A_{n-3}.
I felt that I had to omit A_{-2} and A_{-1}, otherwise the sequence was too easy.
Sequence #10. (moderate). Let p = 314159265, Pi_b be the binary representation of p, read backwards. Then A_n is the n+1th digit of Pi_b.
I tried my best to make sure that these sequences weren't in Sloans, and weren't too much of a Rorschach test. I hope that those of you who did work on the sequences had a fun time. =D
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