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 A339694 Triangle read by rows: A(n, k) = Sum_{i=0..n-1} x(i, k)*2^i, where x(i, k) = A014682^(i)(k) (mod 2) using the i-th iteration of A014682. 2
 0, 1, 0, 1, 2, 3, 0, 5, 2, 3, 4, 1, 6, 7, 0, 5, 10, 3, 4, 1, 6, 7, 8, 13, 2, 11, 12, 9, 14, 15, 0, 21, 10, 3, 20, 17, 6, 23, 8, 29, 2, 11, 12, 9, 14, 15, 16, 5, 26, 19, 4, 1, 22, 7, 24, 13, 18, 27, 28, 25, 30, 31, 0, 21, 42, 35, 20, 17, 6, 23, 40, 29, 34, 11 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS A(n, k) is periodic with period 2^n, i.e., A(n, k) = A(n, k + 2^n). Each row in the triangle is therefore [A(n, 0), A(n, 1), ..., A(n, 2^n-1)]. The binary modular Collatz graph C(n) is the graph representing the dynamics of the Collatz function (A014682) modulo 2^n. For example, in C(3), there is an arrow from 3 to 5 and from 3 to 1 because any number that is 3 modulo 8 either gets mapped to 5 modulo 8 or 1 modulo 8. The vertices of the de Bruijn graph B(2,n) are words of length n consisting of the two symbols 0 and 1. If one represents these vertices as integers, b_0 b_1 ... b_{n-1} -> Sum_{i=0..n-1} b_i*2^i, then A(n) : C(n) -> B(2,n) is a graph isomorphism [Laarhoven, de Weger]. The n-th row is a permutation on the set {0..2^n-1}. For n > 5, the order of this permutation is 2^(n-4) [Bernstein, Lagarias]. - Sebastian Karlsson, Jan 17 2021 LINKS Sebastian Karlsson, Rows n = 1..13, flattened D. J. Bernstein and J. C. Lagarias, The 3x+1 conjugacy map, Canadian Journal of Mathematics, Vol. 48, 1996, pp. 1154-1169. Thijs Laarhoven and Benne de Weger, The Collatz conjecture and De Bruijn graphs, Indagationes Mathematicae. New Series, 24(4) (2013), 971-983. arXiv version, arXiv:1209.3495 [math.NT], 2012. J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23. J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23. FORMULA A000120( T(n, (m + 1) mod 2^n) ) = log_3( A014682^n(m + 1 + 2^n) - A014682^n(m + 1) ), m = 0..2^n-1. (A000120 is the binary weight.) - Thomas Scheuerle, Aug 23 2021 EXAMPLE Triangle begins: n=1 : 0 1; n=2 : 0 1  2 3; n=3 : 0 5  2 3 4 1 6 7; n=4 : 0 5 10 3 4 1 6 7 8 13 2 11 12 9 14 15; ... A(3, 4) = Sum_{i=0..2} x(i, 4)*2^i = 0*2^0 + 0*2^1 + 1*2^2 = 4. A(4, 1) = Sum_{i=0..3} x(i, 1)*2^i = 1*2^0 + 0*2^1 + 1*2^2 + 0*2^3 = 5. PROG (Python) def A014682(k):     if k % 2 == 0:         return k // 2     else:         return (3*k + 1) // 2 def x(i, k):     while i > 0:         k = A014682(k)         i = i - 1     return k % 2 def A(n, k):     L = [x(i, k) * 2**i for i in range(0, n)]     return sum(L) (PARI) f(n) = if(n%2, 3*n+1, n)/2 \\ A014682 x(i, n) = my(x=n); for (k=1, i, x = f(x)); x % 2; A(n, k) = sum(i=0, k-1, x(i, n)*2^i); row(n) = vector(2^n, i, A(i-1, n)); tabf(nn) = for (n=1, nn, print(row(n))); \\ Michel Marcus, Dec 21 2020 CROSSREFS Cf. A014682, A304715. Cf. A000004 (column 0), A052992 (column 1), A263053 (column 2). Sequence in context: A071322 A072594 A272591 * A074722 A331102 A080368 Adjacent sequences:  A339691 A339692 A339693 * A339695 A339696 A339697 KEYWORD nonn,tabf AUTHOR Sebastian Karlsson, Dec 13 2020 STATUS approved

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Last modified January 27 06:18 EST 2022. Contains 350601 sequences. (Running on oeis4.)