Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

Hence, if *n* is the number of 1s,

Now

Thus

Hence we have the required formula

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[/align]*Last edited by JaneFairfax (2009-10-25 01:20:57)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,681

Hi Jane;

Isn't it unfortunate the a_n do not continue a predictable pattern. I know you did not say they did. Just looking at your stuff.

*Last edited by bobbym (2009-10-25 05:53:04)*

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,681

Hi Jane;

A recurrence like this, is similar to a full history recurrence. They are easy to solve:

Theorem: Every full history recurrence can be changed into a finite history recurrence by the method of differences.

Form a new recurrence:

Now subtract them and you have a finite history recurrence.

*Last edited by bobbym (2009-10-25 09:55:40)*

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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**soroban****Member**- Registered: 2007-03-09
- Posts: 452

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