Triangular numbers, T(An +B), that equal (Cn+D)*(En+F) for all n

[ramsey2879]

I found that interdependent arithmetic sequences:
A*n + B, C*n + D, and E*n +F solving the
formula T(A*n+B)=(C*n+D)*(E*n+F), for all integer n can be
generated by the equations
A=(2m+1)*(2m+2)
B=(2m+2)*2m
C=+/- (2m+2)*(m+1)
D=+/- (2m+1)*(m+1)
E=+/- (2m+1)*(2m+1)
F=+/- (2m+1)*2m
I tried to find an example not generated by these formulas but could not. I believe that this finding has many implications with congruences.

 

[ramsey2879]

More on Triangular Numbers and arithmetic sequences

I found two more solution families to T(An+B)=(Cn+D)*(En+F) for all n where T(x) = x*(x+1)/2. Plus there is an interesting interrelation between the solutions. Besides the solutions for A,B,C,D,E and F below there is the solution:
A'=8m*(2m+1)
B'=4m*(3m+1)
C'=(2m+1)*(4m+2)
D'=+/- (2m+1)*(3m+2)
E'=+/- 16m^2
F'=+/- 2m*(6m+1)
Interestingly the above solution and the previous solution merge to form the third solution as follows:
A"=A'
B"=B
C"=C'
D"=D
E"=E'
F"=F
I am further searching for more solution sets.

I found that interdependent arithmetic sequences:
A*n + B, C*n + D, and E*n +F solving the
formula T(A*n+B)=(C*n+D)*(E*n+F), for all integer n can be
generated by the equations
A=(2m+1)*(2m+2)
B=(2m+2)*2m
C=+/- (2m+2)*(m+1)
D=+/- (2m+1)*(m+1)
E=+/- (2m+1)*(2m+1)
F=+/- (2m+1)*2m
I tried to find an example not generated by these formulas but could not. I believe that this finding has many implications with congruences.

 

[ramsey2879]

Group created on Triangular and Figurate Numbers

I posted my further findings and some links at this group which I initiated:
http://groups.yahoo.com/group/Figurate_Numbers/
I present there a general method to find other families of solutions for values of A-F. I have more to post and intend to do so on this new group in the future

 

[HallsofIvy]

Do you realize that you are getting no responses because no one can understand what you are talking about?

What is your point? You start by creating sequences have a certain relationship and then show that the relation can be written in a number of different ways. If you in 9th or 10th grade then, Okay, pretty good. If you are older than that, all that should be trivial to you.