Find a formula for this sequence

[nate9228]

Homework Statement

A sequence (x_j) where j can go from 0 to infinity satisfies the following:
(1) x₁= 1 and
(2) for all m≥n≥0, x_m+n+ x_m-n= (1/2)(x_2m+x_2n)
Find a formula for x_j and prove that the formula is correct

Homework Equations

The Attempt at a Solution

All I have done so far is is play around with m and n values so far and I think a proper formula:
x_j= j². Now I would have to prove this (for which I plan to use induction most likely) but I wanted to see if I was on the write track in my thinking.

 

[Dick]

Homework Statement

A sequence (x_j) where j can go from 0 to infinity satisfies the following:
(1) x₁= 1 and
(2) for all m≥n≥0, x_m+n+ x_m-n= (1/2)(x_2m+x_2n)
Find a formula for x_j and prove that the formula is correct

Homework Equations

The Attempt at a Solution

All I have done so far is is play around with m and n values so far and I think a proper formula:
x_j= j². Now I would have to prove this (for which I plan to use induction most likely) but I wanted to see if I was on the write track in my thinking.

Looks fine to me. Now try and check it. Induction isn't really necessary. Just check that your guess works.