(1, 1.0) (2, 1.618033988749895) (3, 1.8392867552141612) (4, 1.9275619754829254) (5, 1.9659482366454852) (6, 1.9835828434243263) (7, 1.991964196605035) (8, 1.9960311797354147) (9, 1.9980294702622867) (10, 1.9990186327101012) (11, 1.9995104019782854) (12, 1.9997555009373176) (13, 1.9998778327115456) (14, 1.9999389387495947) (15, 1.9999694754345032) (16, 1.9999847393479442) (17, 1.9999923701106546) (18, 1.9999961851717603) (19, 1.9999980926168055) (20, 1.9999990463165884) (21, 1.9999995231604544) (22, 1.9999997615807956) (23, 1.999999880790547) (24, 1.9999999403953126) (25, 1.9999999701976665) (26, 1.999999985098836) (27, 1.9999999925494187) (28, 1.9999999962747095) (29, 1.9999999981373549) (30, 1.9999999990686774) (31, 1.9999999995343387) (32, 1.9999999997671694) (33, 1.9999999998835847) (34, 1.9999999999417923) (35, 1.9999999999708962) (36, 1.999999999985448) (37, 1.999999999992724) (38, 1.999999999996362) (39, 1.999999999998181) (40, 1.9999999999990905) (41, 1.9999999999995453) (42, 1.9999999999997726) (43, 1.9999999999998863) (44, 1.9999999999999432) (45, 1.9999999999999716) (46, 1.9999999999999858) (47, 1.999999999999993) (48, 1.9999999999999964) (49, 1.9999999999999982)
I ran this on a computer. The first member of each of the above ordered pairs represents the number of initial elements in the sequence, i.e., $n$. For example, when $n = 2$,
Initial sequence = ${1, 1}$
We add the last $n$ terms of the sequence to produce the next term.
${1, 1, 2} \\ {1, 1, 2, 3} \\ {1, 1, 2, 3, 5}, etc.$
When $n = 3$
${1, 1, 1} \\ {1, 1, 1, 3} \\ {1, 1, 1, 3, 5} \\ {1, 1, 1, 3, 5, 9}, etc.$
The ratio of adjacent numbers in these sequences all approach a certain limit. The limits for each $n$ from 1 to 49 are quoted above. It is well known that the ratio of adjacent terms in the 2-bonacci (Fibonacci) sequence tends to $\phi = \frac{1 + \sqrt{5}}{2}$.
It doesn't approach two itself. Each n-bonacci sequence has an adjacent-term-ratio limiting to some value less than two. For instance, I ran 49-bonacci up to thousand and then ten thousand terms and the ratio of the last two numbers was 1.9999999999999982 in both cases (low-res computer). But as the value $n$ gets bigger, the limit itself gets closer and closer to two. For instance, limit of 25-bonacci is 1.9999999701976665, whereas that of 30-bonacci is 1.9999999990686774.
Why is it so?
