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A375018
Numbers k such that repeated application of the Pisano period eventually gives 24.
0
2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 17, 18, 19, 21, 23, 24, 26, 27, 28, 29, 32, 34, 36, 37, 38, 39, 42, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 63, 64, 67, 68, 69, 72, 73, 74, 76, 78, 79, 81, 83, 84, 87, 91, 92, 94, 96, 97, 98
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.
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
Cf. A001175.
Sequence in context: A285986 A101883 A236207 * A225837 A035246 A356955
KEYWORD
nonn
AUTHOR
STATUS
approved