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%I #13 Nov 05 2015 14:31:22
%S 1,5,1,9,17,13,5,1,13,5,1,17,13,5,1,21,1,25,29,17,13,5,1,29,17,13,5,1,
%T 33,25,29,17,13,5,1,37,17,13,5,1,41,161,121,137,233,593,445,377,425,
%U 2429,3077,577,433,325,61,53,5,1,45,17,13,5,1
%N Irregular triangle T read by rows in which the entry in row n and column k is given by T(n,k) = 4*A253676(n,k) - 3, k = 1..A253720(n), assuming the 3x+1 (or Collatz) conjecture.
%C Definitions: Let v(y) denote the 2-adic valuation of y. Let N_1 denote the set of odd natural numbers. Let F : N_1 -> N_1 be the map defined by F(x) = (3*x + 1)/2^v(3*x + 1) (cf. A075677). Let F^(k)(x) denote k-fold iteration of F and defined by the recurrence F^(k)(x) = F(F^(k-1)(x)), k>0, with initial condition F^(0)(x) = x.
%C This triangle can be constructed by restricting the initial values to the numbers 4*n - 3, iterating F until 1 is reached (assuming the 3x+1 conjecture) and removing all iterates not congruent to 1 modulo 4. Equivalently, for each n, this is accomplished by iterating (until 1 is reached, assuming the 3x+1 conjecture) the function S defined in A257480 to get the triangle A253676, and finally taking T(n,k) = 4*A253676(n,k) - 3.
%C Conjecture: For each natural number n, there exists a k >= 0, such that F^k(4*n - 3) = 1.
%C Theorem 1: Conjecture 1 is equivalent to the 3x+1 (or Collatz) conjecture.
%C Proof: See A257480.
%e T begins:
%e . 1
%e . 5 1
%e . 9 17 13 5 1
%e . 13 5 1
%e . 17 13 5 1
%e . 21 1
%e . 25 29 17 13 5 1
%e . 29 17 13 5 1
%e . 33 25 29 17 13 5 1
%e . 37 17 13 5 1
%e . 41 161 121 137 233 593 445 377 425 2429 3077 577 433 325 61 53 5 1
%e . 45 17 13 5 1
%e . 49 37 17 13 5 1
%e . 53 5 1
%e . 57 65 49 37 17 13 5 1}
%t v[x_] := IntegerExponent[x, 2]; f[x_] := (3*x + 1)/2^v[3*x + 1]; s[n_] := NestWhileList[(3 + (3/2)^v[1 + f[4*# - 3]]*(1 + f[4*# - 3]))/6 &, n, # > 1 &]; t = Table[4*s[n] - 3, {n, 1, 15}]; Flatten[t] (* Replace Flatten with Grid to display the triangle *)
%Y Cf. A014682, A070165, A075677, A256598, A257480, A253676, A254070.
%K nonn,tabf
%O 1,2
%A _L. Edson Jeffery_, May 03 2015
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