Help extending equation -- sequence related to pentagonal numbers

[Marc LeBrun]

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!