Restrictions for rule of general sequence: $T_{n+2}=T_{n} + T_{n+1}$

Question

This is a rule which gives a sequence where $11$ times the seventh term ($11 t_7$) is equal to the sum of the first $10$ terms ($s_{10}$), where the first two starting numbers can be chosen. Do any restrictions exist on the first two starting numbers? And what's an example that would validate this?

Answer 1

If the first two terms are $a$ and $b$ then you can add up terms until you get to the 7th. It will be:

$$t_7=8a+13b$$

Similarly you can write down and add up the first 10 terms to get:

$$s_{10}=\sum_{n=1}^{10}t_n=88a+143b$$

Solving the equality now gives:

$$11(8a+13b)=88a+143b$$

As both sides are equal the first two numbers can be anything.

Answer 2

If one denotes the first terms by $a_0$ and $a_1$, the $7$th term will be a linear combination of those, say $p_7 a_0+ q_7 a_1$. Same goes for the sum: $s_{10} =u_{10} a_0+ v_{10} a_1$. You do not really need to compute explicitely those linear terms.

If you choose peculiar solutions: $a_0=0$ and $a_1=1$, the property works ($t^{0,1}_7=8$, $s^{0,1}_{10}=88$). If you choose $a_0=1$ and $a_1=0$, the property works as well ($t^{1,0}_7=5$, $s^{1,0}_{10}=55$).

Now for any $a_0$ and $a_1$, by linear combination of the above solutions, this will work too. More precisely, $s_{10} =s^{1,0}_{10} a_0+ s^{0,1}_{10} a_1$, similarly for $t_7$.

The two simple solutions, seen as vectors, provide a basis of the generic solutions. This could be handy if you look for similar equations with much bigger indices.