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What is the recurrence relation for 1, 7, 31, 127, 499?

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What is the recurrence relation for 1, 7, 31, 127, 499?

(a) bn+1=5bn-1+3

(b) bn=4bn+7!

(c) bn=4bn-1+3

(d) bn=bn-1+1

This question was addressed to me during an interview for a job.

The doubt is from Advanced Counting Techniques in section Counting of Discrete Mathematics

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Right option is (c) bn=4bn-1+3

For explanation: Look at the differences between terms: 1, 7, 31, 124,…. and these are growing by a factor of 4. So, 1⋅4=4, 7⋅4=28, 31⋅4=124, and so on. Note that we always end up with 3 less than the next term. So, bn=4bn-1+3 is the recurrence relation and the initial condition is b0=1.

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