Programming a recursion

[AlfredFlaßhaar]

A friendly hello to all,
again i need help. It's about the recursion u(n+2)=7*u(n+1)-3*u(n)+3 for natural numbers n. How does the calculation and tabulation work in MC15? In derive I managed it successfully. But in MC15 everything is possible, just not the correct result. I just can't get a grip on programming in MC 😞 .
The background is the transformation of the formula u(n)=((1/2)*(7+sqrt(37))^n+((1/2)*(7-sqrt(37))^n-1 given in Binet-Art into a recursion and examination of divisibility by 3.
Kind regards, Alfred Flasshaar

[Werner_E]

What about the initial values u(0) and u(1) ?

[AlfredFlaßhaar]

The Binet formula given here and the recursion are equivalent, which was derived in a different way (characteristic polynomial). Accordingly, u(1)=6 and u(2)=42.

[Werner_E · Accepted solution]

See if the attached file helps.

I wonder why you would like to use the recursion when a non-recursive formula is available.

[AlfredFlaßhaar]

The proof that the Binet term is divisible by 3 becomes almost obvious in the recursion when inductive theory is applied. This got me interested after I also learned about a direct tricky short proof.
Thank you, the MC file is instructive for me.