[seqfan] Re: 2*x+3*y<=n

franktaw at netscape.net franktaw at netscape.net
Wed Oct 7 07:06:17 CEST 2009

Even more directly, the end of the first comment is "the number of 
nonnegative solutions to b+2c+3d=n".  n-b is an arbitrary number <= n, 
and c and d are your x and y.

Franklin T. Adams-Watters

-----Original Message-----
From: N. J. A. Sloane <njas at research.att.com>

Or more directly, if 2x+3y <= n, let w pick up
the slack, and write 1*w + 2*x +3*y = n.
This is a partition of n into parts of size <= 3,
which matches one of the definitions of A001399

>From seqfan-bounces at list.seqfan.eu Tue Oct  6 23:08:24 2009
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>Date: Tue, 6 Oct 2009 20:03:02 -0700 (PDT)
>From: Robert Israel <israel at math.ubc.ca>
>To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>Subject: [seqfan] Re: 2*x+3*y<=n
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>Let a(n) be the number of nonnegative x,y integer solutions to
>2*x + 3*y <= n.  Each solution [x,y] is either [0, y] with 3y <= n
>divisible by 3 or [1+t,y] where [t,y] is a solution to
>2*t + 3*y <= n-2.  So a(n) = a(n-1) + floor(n/3). This
>corresponds to one of the entries in the FORMULA section:
>a(n) = a(n-2)+A008620(n).

>Robert Israel                                israel at math.ubc.ca
>Department of Mathematics        http://www.math.ubc.ca/~israel
>University of British Columbia            Vancouver, BC, Canada

>On Wed, 7 Oct 2009, rhhardin at att.net wrote:

>> Is it obvious from the comments to
>> http://www.research.att.com/~njas/sequences/A001399
>> 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 30, 33, 37, 
40, 44,
48, 52
>> that it's the number of nonnegative x,y integer solutions to
>> 2*x+3*y <= n
>> (empirical)
>> --
>> rhhardin at mindspring.com
>> rhhardin at att.net (either)

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