# Different upper and lower bound

If [math]N(a,b)[/math] is the maximum length of a [math]\pm 1[/math] sequence with the partial sums along its HAPs bounded below by [math]-a[/math] and above by [math]b[/math], then:

[math]N(a, b) = N(b, a)[/math]

[math]N(0, b) = b[/math] (everything must be [math]+1[/math])

[math]N(1, 1) = 11[/math] (there are [math]4[/math] such sequences: choose [math]x_1[/math], and use the constraints [math]x_n +x_{2n} = 0[/math] and [math]x_1 + \ldots + x_{2n} = 0[/math] to determine the entries up to [math]10[/math]; then choose [math]x_{11}[/math]) (alternatively: Our first placement can be arbitrary, so suppose 6 is +. This forces 12-, which forces 3- and 9+, which forces 2- and 4+, which forces 1+ and 8-, which forces 5- and 10+, which forces 7+, and a contradiction when d = 1 and k = 10. The sequence is thus: + – - + – + + – + + . -)

[math]N(1, 2) = 41[/math] (there are [math]4[/math] such sequences -- example below)

[math]N(1, 3) = 83[/math] (there are [math]216[/math] such sequences -- example below)

[math]N(1, 4) = 131[/math] (there are [math]87144[/math] such sequences -- example below)

[math]N(1,b)\lt\infty[/math] (On a conjecture of Erdős and Čudakov)

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Question: is [math]N(1,b)[/math] always prime?
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[math]N(2, 2) = 1160[/math] (e.g the first 1124-sequence)

For the zero-based problem, see this comment

## Contents

## Method

Here should be a short description of the way the sequences was found. (The code(s) used should be further down this page.)

## Status

Is the data still relevant (e.g. longest know)? Is the method still relevant, or have we found a better method? Is the program still running on a computer somewhere?

## The data

[math]N(1, 2) = 41[/math] (there are [math]4[/math] such sequences -- example below)

0 + - - + - + + - + + - - + - + + - + - - - + - + + - - + + - + - + + - - - + - + +

[math]N(1, 3) = 83[/math] (there are [math]216[/math] such sequences -- example below)

0 - + - + - + + - + + - - + - + + - - + - - + - + + - + + + - - - + + - + + - - + - + - - + + - - + - + + + - + - - - + + - + - + - - + - + + - + + - - + - + - - - + +

[math]N(1, 4) = 131[/math] (there are [math]87144[/math] such sequences -- example below)

0 + - - + - + + - - + - + + - + + + + - - - + - - + - + + - - + + + - - - + + - + - + - - + + - + + - - + - - + - + + + + - - + - + - - + + + - + - - - - + + - - - + + - - + + + + - - - - + + - + - + + + + - - + + - + - - - + + - - - + - - + + + - + + - - + - - +

[math]N(2, 2) = 1160[/math]

--Alec 13:46, 9 January 2010 (UTC)

## Relevant code

The code(s) (or a link to the code(s)) used to find this sequence should be posted here.