OFFSET
1,1
COMMENTS
This sequence is infinite. A number n is a fixed point if the Pisano period of n is equal to n. The trajectory of k is the sequence of values the Pisano period takes on under repeated iteration, starting at k and leading to a fixed point; this sequence is the sequence of integers such that the trajectory leads to 24.
LINKS
B. Benfield and O. Lippard, Fixed points of K-Fibonacci Pisano periods, arXiv:2404.08194 [math.NT], 2024.
J. Fulton and W. Morris, On arithmetical functions related to the Fibonacci numbers, Acta Arith., 2(16):105-110, 1969.
E. Trojovska, On periodic points of the order of appearance in the Fibonacci sequence, Mathematics, 2020.
EXAMPLE
a(1)=2 because 2 is the smallest number with Pisano period trajectory terminating at 24: pi(2)=3, pi(3)=8, pi(8)=12, pi(12)=24.
PROG
(Sage)
L=[]
for i in range(2, 101):
a=i
y=BinaryRecurrenceSequence(1, 1, 0, 1).period(Integer(i))
while a!=y:
a=y
y=BinaryRecurrenceSequence(1, 1, 0, 1).period(Integer(a))
if a==24:
L.append(i)
print(L)
CROSSREFS
KEYWORD
nonn
AUTHOR
Oliver Lippard and Brennan G. Benfield, Aug 04 2024
STATUS
approved