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wu :: forums    riddles    medium (Moderators: towr, Icarus, Grimbal, Eigenray, ThudnBlunder, SMQ, william wu)    An Interesting Recursion « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: An Interesting Recursion  (Read 639 times) ThudnBlunder wu::riddles Moderator Uberpuzzler The dewdrop slides into the shining Sea     Gender: Posts: 4489 An Interesting Recursion   « on: Nov 26th, 2008, 9:30am » Quote Modify Consider the recursive formula a(n) = a(a(n-1)) + a(n - a(n-1))   1) What is a(3)?   2) When does a(n)/n equal 0.5?   3) What is the least value of n such that a(n)/n - 0.5 < 0.05 for all greater n? IP Logged THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you. towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: An Interesting Recursion   « Reply #1 on: Nov 26th, 2008, 10:08am » Quote Modify For a(n)=0 1) 0 2) never,   3) doesn't exist IP Logged Wikipedia, Google, Mathworld, Integer sequence DB SMQ wu::riddles Moderator Uberpuzzler     Gender: Posts: 2084 Re: An Interesting Recursion   « Reply #2 on: Nov 26th, 2008, 10:12am » Quote Modify What's the base case for the recursion?   My first thoughts are that in order for the recursion to be evaluated straightforwardly, 0 < a(n) <= n.  If a(n-1) <= 0 for some n, then n - a(n-1) >= n, and a(n - a(n-1)) is unknown; likewise if a(n-1) > n-1 for some n then a(a(n-1)) is unknown.   Moreover, assuming the field of integers, 0 < a(n) <= n implies a(1) = 1; but if a(1) = 1, than a(2) = 2 and a(n) = a(n-1) + a(1) = n for all positive integer n.   --SMQ IP Logged --SMQ ThudnBlunder wu::riddles Moderator Uberpuzzler The dewdrop slides into the shining Sea     Gender: Posts: 4489 Re: An Interesting Recursion   « Reply #3 on: Nov 26th, 2008, 10:14am » Quote Modify on Nov 26th, 2008, 10:08am, towr wrote: For a(n)=0 Oh yes, a(1) = a(2) = 1   I always forget something.   IP Logged THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you. towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: An Interesting Recursion   « Reply #4 on: Nov 26th, 2008, 10:28am » Quote Modify Ah, well, in that case. 1) 2 2) for any n=2k 3) no idea there « Last Edit: Nov 26th, 2008, 10:28am by towr » IP Logged Wikipedia, Google, Mathworld, Integer sequence DB ThudnBlunder wu::riddles Moderator Uberpuzzler The dewdrop slides into the shining Sea     Gender: Posts: 4489 Re: An Interesting Recursion   « Reply #5 on: Dec 4th, 2008, 4:09pm » Quote Modify Mathworld has a page about this batrachion. IP Logged THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you. Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------  - easy => medium   - hard   - what am i   - what happened   - microsoft   - cs   - putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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