Who can show the first 20 terms of a Hexanacci sequence?

Question

Hint: the Fibonacci Sequence actually starts with zero (0) not one (1).

Answer 1

The hexanacci numbers are a generalization of the Fibonacci numbers defined by , , , , , , and the recurrence relation

(1)

for . They represent the case of the Fibonacci n-step numbers.

The first few terms for , 2, ... are 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, ... (Sloane's A001592).

An exact formula for the th heptanacci number can be given explicitly in terms of the six roots of

(2)

as

(3)

The ratio of adjacent terms tends to the positive root of , namely 1.98358284342... (Sloane's A118427), sometimes called the hexanacci constant.