OFFSET
1,3
COMMENTS
The length of the rows is given by A002326. For n > 1, the first term of row n is 1 and the last term is n. Many sequences are in this one: starting at A036117 (mod 11) and A070335 (mod 23).
Row n, for n >= 2, divided elementwise by (2*n-1) gives the cycles of iterations of the doubling function D(x) = 2*x or 2*x-1 if 0 <= x < 1/2 or , 1/2 <= x < 1, respectively, with seed 1/(2*n-1). See the Devaney reference, pp. 25-26. D^[k](x) = frac(2^k/(2*n-1)), for k = 0, 1, ..., A002326(n-1) - 1. E.g., n = 3: 1/5, 2/5, 4/5, 3/5. - Gary W. Adamson and Wolfdieter Lang, Jul 29 2020.
REFERENCES
Robert L. Devaney, A First Course in Chaotic Dynamical Systems, Addison-Wesley., 1992. pp. 24-25
LINKS
T. D. Noe, Rows n = 1..100, flattened
FORMULA
T(n, k) = 2^k mod (2*n-1), n >= 1, k = 0, 1, ..., A002326(n-1) - 1.
T(n, k) = (2*n-1)*frac(2^k/(2*n-1)), n >= 1, k = 0, 1, ..., A002326(n-1) - 1, with the fractional part frac(x) = x - floor(x). - Wolfdieter Lang, Jul 29 2020
EXAMPLE
The irregular triangle T(n, k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ...
---------------------------------------------------------
1: 0
2: 1 2
3: 1 2 4 3
4: 1 2 4
5: 1 2 4 8 7 5
6: 1 2 4 8 5 10 9 7 3 6
7: 1 2 4 8 3 6 12 11 9 5 10 7
8: 1 2 4 8
9: 1 2 4 8 16 15 13 9
10: 1 2 4 8 16 13 7 14 9 18 17 15 11 3 6 12 5 10
... reformatted by Wolfdieter Lang, Jul 29 2020.
MATHEMATICA
nn = 30; p = 2; t = p^Range[0, nn]; Flatten[Table[If[IntegerQ[Log[p, n]], {0}, tm = Mod[t, n]; len = Position[tm, 1, 1, 2][[-1, 1]]; Take[tm, len-1]], {n, 1, nn, 2}]]
PROG
(GAP) R:=List([0..72], n->OrderMod(2, 2*n+1));;
Flat(Concatenation([0], List([2..11], n->List([0..R[n]-1], k->PowerMod(2, k, 2*n-1))))); # Muniru A Asiru, Feb 02 2019
CROSSREFS
Cf. A000034 (3), A070402 (5), A069705 (7), A036117 (11), A036118 (13), A062116 (17), A036120 (19), A070347 (21), A070335 (23), A070336 (25), A070337 (27), A036122 (29), A070338 (33), A070339 (35), A036124 (37), A070340 (39), A070348 (41), A070349 (43), A070350 (45), A070351 (47), A036128 (53), A036129 (59), A036130 (61), A036131 (67), A036135 (83), A036138 (101), A036140 (107), A201920 (125), A036144 (131), A036146 (139), A036147 (149), A036150 (163), A036152 (173), A036153 (179), A036154 (181), A036157 (197), A036159 (211), A036161 (227).
KEYWORD
nonn,easy,tabf
AUTHOR
T. D. Noe, Dec 07 2011
STATUS
approved