Help extending equation -- sequence related to pentagonal numbers

[Andrew Plewe]

Point taken concerning the algebra -- I'll work it out.  I'd like to come up
with a generalized equation similar to that for A005449, or (n(3n+1))/2,
which will give all values for the desired sequence with f(x), x = 0 as the
"starting" point.  Incidentally, I find these sequences interesting because
I ran into them while doing the following simple transformation (add the
first two terms and redistribute over the remaining terms, then sum the
result):

1 + 1 + 2+3 = 3+4 = 7
1 + 2 + 3+4+5 = 4+5+6 = 15
1 + 3 + 4+5+6+7 = 5+6+7+8 = 26

which can be described by 7 + 8(x) + (3 * tri(x-1), starting with x = 0.

And:

2 + 1 + 2+3+4 = 3+4+5 = 12
2 + 2 + 3+4+5+6 = 4+5+6+7 = 22
2 + 3 + 4+5+6+7+8 = 5+6+7+8+9 = 35

which can be described by  12 + 10(x) + (3 * tri(x-1)), starting with x = 0.

Anyway, thanks for you help!

	-Andrew Plewe-

-----Original Message-----
From: Marc LeBrun [mailto:mlb at fxpt.com]
Sent: Thursday, December 22, 2005 7:03 PM
To: seqfan
Subject: Re: Help extending equation -- sequence related to pentagonal
numbers

I'm not 100% sure what you are asking, but maybe this will help:

You have a bunch of statements of the form

   "Equation to produce sequence from Kth term: u + v(x) + (3 * tri(x-1))"

So let's restate this using a general parameterized formula we'll define:
f[u,v](x) = u + v x + 3 tri(x-1).

(You usually make this statement for K=4--except the first time, when you
use K=3 for some reason--but no matter...)

You give a few "starting values" and then you're basically saying that
everything after that is f(1), f(2), f(3) and so on.

But, at least for all those I checked, your formula applied to x<1, that is
f(0), f(-1), f(-2) and so on, seems to give the "starting values" as well.

So I'll leap to the conclusion that what you want is the formula, a(N)
defined so that it that "starts" at the first term (ie N=1), rather than f
which "starts" at the Kth term.

This is easy: because a(K)=f(1), a(K+1)=f(2), and so on, it's simply an
offset, so a(N)=f(N-K+1).

So now all we have to do is substitute x=N-K-1 to get the formula for a(N):

   a(N) = f(N-K+1) = u + v (N-K-1) + 3 tri(N-K-2)

I'll leave simplifying the algebra to you.

By the way, I remark that these a(N) are just quadratics in N, and so their
properties are well understood (eg 3-term recurrence, rational 2nd-degree
generating functions, etc), so you probably should only trouble to submit
them if they have some other especially interesting property (such as being
pentagonal or whatever).

========

At 05:14 PM 12/22/2005, Andrew Plewe wrote:
>My new sequence follows from extending a group of partial equations for
>other sequences related to pentagonal/triangle numbers in the OEIS.  To
>demonstrate the pattern, I've listed the sequences in the OEIS with
>their corresponding partial generating equations.  I'd like to find a.)
>an equation that fully describes my new sequence, and b.) a general
>method for deriving equations for other similar sequences. "tri", here,
>represents the function for generating triangle numbers ((n * (n + 1))/2:
>
>A005449 = 0, 2, 7, 15, 26, 40, etc.  Equation to produce sequence from
>third term: 7 + 8(x) + (3 * tri(x-1))
>
>A000326 = 0, 1, 5, 12, 22, 35, etc.  Equation to produce sequence from
>fourth term: 12 + 10(x) + (3 * tri(x-1))
>
>A045943 = 0, 3, 9, 18, 30, 45, etc.  Equation to produce sequence from
>fourth term: 18 + 12(x) + (3 * tri(x-1))
>
>A095794 = 1, 6, 14, 25, 39, 56, etc.  Equation to produce sequence from
>fourth term: 25 + 14(x) + (3 * tri(x-1))
>
>And my new sequence:
>
>A000001 = 3, 10, 20, 33, 49, 68, etc.  Equation to produce sequence
>from fourth term: 33 + 16(x) + (3 * tri(x-1))
>
>
>As you can see there is a pattern to these generating equations.
>Looking at the equations already in the database for the sequences
>listed above, I was unable to determine a general method for generating
>equations which would fully describe sequences derived in this manner.
>I'm not sure if it's o.k. to submit equations that partially describe a
>sequence to the OEIS, so I'd like to find one which fully describes my
>sequence before submitting it.  Any help is appreciated.  Thanks!