# [seqfan] Re: Period lengths of k^2 mod n not in OEIS ?

Richard Mathar mathar at strw.leidenuniv.nl
Fri Mar 11 13:57:44 CET 2011

```followup on
http://list.seqfan.eu/pipermail/seqfan/2011-February/007219.html

The coverage of Pisano period lengths in the OEIS is -- complete to the best
of my knowledge--:

A000045 A001175          [1, 1]    [1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, 24, 28, 48, 40, 24, 36, 24, 18, 60]
A000073 A046738       [1, 1, 1]    [1, 4, 13, 8, 31, 52, 48, 16, 39, 124, 110, 104, 168, 48, 403, 32, 96, 156, 360, 248]
A000129 A175181          [2, 1]    [1, 2, 8, 4, 12, 8, 6, 8, 24, 12, 24, 8, 28, 6, 24, 16, 16, 24, 40, 12]
A000290 A186646      [3, -3, 1]  m [1, 2, 3, 2, 5, 6, 7, 4, 9, 10, 11, 6, 13, 14, 15, 8, 17, 18, 19, 10]
A000578          [4, -6, 4, -1]  m [1, 2, 3, 4, 5, 6, 7, 8, 3, 10, 11, 12, 13, 14, 15, 16, 17, 6, 19, 20]
A001045 A175286          [1, 2]    [1, 1, 6, 2, 4, 6, 6, 2, 18, 4, 10, 6, 12, 6, 12, 2, 8, 18, 18, 4]
A001076 A175183          [4, 1]    [1, 2, 8, 2, 20, 8, 16, 4, 8, 20, 10, 8, 28, 16, 40, 8, 12, 8, 6, 20]
A001608 A104217       [0, 1, 1]    [1, 7, 13, 14, 24, 91, 48, 28, 39, 168, 120, 182, 183, 336, 312, 56, 288, 273, 180, 168]
A001644 A106293       [1, 1, 1]  m [1, 1, 13, 4, 31, 13, 48, 8, 39, 31, 10, 52, 168, 48, 403, 16, 96, 39, 360, 124]
A002605 A175289          [2, 2]    [1, 1, 3, 1, 24, 3, 48, 1, 9, 24, 10, 3, 12, 48, 24, 1, 144, 9, 180, 24]
A005668 A175185          [6, 1]    [1, 2, 2, 4, 20, 2, 16, 8, 6, 20, 24, 4, 6, 16, 20, 16, 36, 6, 8, 20]
A006130 A175291          [1, 3]    [1, 3, 1, 6, 24, 3, 24, 6, 3, 24, 120, 6, 156, 24, 24, 12, 16, 3, 90, 24]
A006190 A175182          [3, 1]    [1, 3, 2, 6, 12, 6, 16, 12, 6, 12, 8, 6, 52, 48, 12, 24, 16, 6, 40, 12]
A030195 A175290          [3, 3]    [1, 3, 1, 3, 4, 3, 42, 6, 1, 12, 120, 3, 84, 42, 4, 12, 16, 3, 90, 12]
A052918 A175184          [5, 1]    [1, 3, 8, 6, 2, 24, 6, 12, 8, 6, 24, 24, 12, 6, 8, 24, 36, 24, 40, 6]

The first A-number is the base sequence, the second A-number (if known) is the
sequence with the lengths of its Pisano periods, then there is is a bracketed
list of integers with the "signature" of the linear recurrence, then there
may be a "m" indicating that the Pisano sequence appears to be multiplicative
(including false positives like A106293, which is not multiplicative),
then there is a bracketed list of the first 20 terms of the Pisano period
lengths, offset 0, which ought to be the start of the sequence with the 2nd
A-number.

Given that each sequence which obeys a linear recurrence with constant
coefficients has Pisano  periods, this list is very small in comparison with
the number of sequences in the category of linear recurrences, estimated to
be some 2000 in the OEIS.

The interesting questions are (perhaps just revealing my lack of knowledge
in that area):
i) which conditions in the base sequence lead to the same sequence of the Pisano period lengths?
ii) which conditions in the base sequence result in multiplicative sequences of period lengths?

Richard J. Mathar

```