A332434 - OEIS

%I #26 Dec 20 2024 09:26:35

%S 1,1,2,1,3,3,2,1,3,5,2,6,5,5,7,2,2,4,1,3,6,9,6,3,7,3,3,5,6,12,10,4,4,

%T 13,10,3,5,15,15,2,4,4,1,3,3,5,17,10,18,2,6,4,6,10,14,20,13,21,2,4,6,

%U 4,14,4,6,4,8,6,4,8,4,4,6,18,11,13,9,7,25,26,4,8,27,9,7,11,18,5,7,9,7,22,4,6,8

%N Irregular triangle read by rows: r-tuples (lengths) of the complete coach system Sigma(2*n+1), for n >= 1.

%C The length of row n of this irregular triangle is A135303(n). The row sums are given in A332435, where more details are found.

%C For the complete coach system Sigma(b), with b = 2*n+1, for n >= 1, see the Hilton and Pedersen [HP] reference. The coach numbers are c(b) = c(2*n+1) = A135303(n), and the quasi-order of 2 modulo b is k(n) = A003558(n). The number of entries (length) of a specific coach of Sigma(b), say C(b; j), for j from {1, 2, ..., c(b)} is r(b;j), and the present array lists the r-tuples R(b) = (r(b; 1), ..., (b; c(b)). These R(b) numbers give the length of the (primitive) periods of the cycles of the first rows of each coach.

%C The parity of the entries of each row is identical ([HP], p. 261).

%C This table and a computation shows that part two of item (2) of 'Some open questions' of [HP], p. 281, namely 'Is it the case that the smallest r always occurs in the first coach (where a_1 =1)?' has the answer no. For the first counterexamples see: b = 46, 99, 109, 155, 157, 189, ..., with the r-tuples (6,4,8), (9,7), (9,7,11), (10,8,12), (13,11,15) (10,8,8), ...

%D Peter Hilton and Jean Pedersen, A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics, Cambridge University Press, 2010, (3rd printing 2012) pp. 261-281.

%H Wolfdieter Lang, <a href="https://arxiv.org/abs/2008.04300">On the Equivalence of Three Complete Cyclic Systems of Integers</a>, arXiv:2008.04300 [math.NT], 2020.

%F T(n, j) gives the length of the j-th coach of the complete coach system Sigma(b), with b = 2*n+1, for n >= 1, and j = 1, 2, ..., A135303(n).

%e The irregular triangle T(n, j) begins:

%e   n,   b \ j   1  2  3 ... | A135303(n)   A332435(n)

%e   1,   3:      1                1            1

%e   2,   5:      1                1            1

%e   3,   7:      2                1            2

%e   4,   9:      1                1            1

%e   5,  11:      3                1            3

%e   6,  13:      3                1            3

%e   7,  15:      2                1            2

%e   8,  17:      1  3             2            4

%e   9,  19:      5                1            5

%e  10,  21:      2                1            2

%e  11,  23:      6                1            6

%e  12,  25:      5                1            5

%e  13,  27:      5                1            5

%e  14,  29:      7                1            7

%e  15,  31:      2  2  4          3            8

%e  16,  33:      1  3             2            4

%e  17,  35:      6                1            6

%e  18,  37:      9                1            9

%e  19,  39:      6                1            6

%e  20,  41:      3  7             2           10

%e  ...

%e In the following the complete coach is written as a list of list of coaches, and the first and second rows (the a- and k-numbers) of a coach are separated by a semicolon. Here only the first part of a coach list (the top row of a coach) is of interest.

%e n = 5, b = 11: Sigma(11) = [[1, 5, 3;  1, 1, 3]], hence T(5, 1) = 3 or R(11) = (r(11,1)) = (3).

%e n = 8, b = 17: Sigma(17) = [[1; 4], [3, 7, 5; 1, 1, 2]], hence T(8, 1) = 1, T(8, 2) = 3.

%e n = 16, b = 33:  Sigma(33) = [[1; 5], [5, 7, 13; 2, 1, 2]], hence T(16, 1) = 1, T(16, 2) = 3.

%Y Cf. A003558, A135303, A332435.

%K nonn,tabf,easy

%O 1,3

%A _Wolfdieter Lang_, Feb 26 2020