A070168 Irregular triangle of Terras-modified Collatz problem.
1, 2, 1, 3, 5, 8, 4, 2, 1, 4, 2, 1, 5, 8, 4, 2, 1, 6, 3, 5, 8, 4, 2, 1, 7, 11, 17, 26, 13, 20, 10, 5, 8, 4, 2, 1, 8, 4, 2, 1, 9, 14, 7, 11, 17, 26, 13, 20, 10, 5, 8, 4, 2, 1, 10, 5, 8, 4, 2, 1, 11, 17, 26, 13, 20, 10, 5, 8, 4, 2, 1, 12, 6, 3, 5, 8, 4, 2, 1, 13, 20, 10, 5, 8, 4, 2, 1, 14, 7, 11
Offset: 1
Examples
The irregular triangle begins: n\k 0 1 2 3 4 5 6 8 9 10 11 12 13 14 ... 1: 1 2: 2 1 3: 3 5 8 4 2 1 4: 4 2 1 5: 5 8 4 2 1 6: 6 3 5 8 4 2 1 7: 7 11 17 26 13 20 10 5 8 4 2 1 8: 8 4 2 1 9: 9 14 7 11 17 26 13 20 10 5 8 4 2 1 10: 10 5 8 4 2 1 11: 11 17 26 13 20 10 5 8 4 2 1 12: 12 6 3 5 8 4 2 1 13: 13 20 10 5 8 4 2 1 14: 14 7 11 17 26 13 20 10 5 8 4 2 1 15: 15 23 35 53 80 40 20 10 5 8 4 2 1 ... formatted by _Wolfdieter Lang_, Mar 20 2014 -------------------------------------------------------------
Links
- Reinhard Zumkeller, Rows n = 1..250 of triangle, flattened
- J. C. Lagarias, The 3x+1 Problem and its Generalizations, Amer. Math. Monthly 92 (1985) 3-23.
- R. Terras, A stopping time problem on the positive integers, Acta Arith. 30 (1976) 241-252.
- Eric Weisstein's World of Mathematics, Collatz Problem
- Wikipedia, Collatz conjecture
Programs
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Haskell
a070168 n k = a070168_tabf !! (n-1) !! (k-1) a070168_tabf = map a070168_row [1..] a070168_row n = (takeWhile (/= 1) $ iterate a014682 n) ++ [1] a070168_list = concat a070168_tabf -- Reinhard Zumkeller, Oct 03 2014
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Mathematica
f[n_] := If[EvenQ[n], n/2, (3 n + 1)/2]; Table[NestWhileList[f, n, # != 1 &], {n, 1, 30}] // Grid (* Geoffrey Critzer, Oct 18 2014 *) -
Python
def a(n): if n==1: return [1] l=[n, ] while True: if n%2==0: n//=2 else: n = (3*n + 1)//2 l.append(n) if n<2: break return l for n in range(1, 16): print(a(n)) # Indranil Ghosh, Apr 15 2017
Formula
From Wolfdieter Lang, Mar 20 2014: (Start)
See Lagarias, pp. 4-7, eqs. (2.1), (2.4) with (2.5) and (2.6).
T(n,k) = T^{(k)}(n), with the iterations of the Terras-modified Collatz map: T(n) = n/2 if n is even and otherwise (3*n+1)/2, n >= 1. T^{(0)}(n) = n.
T(n,k) = lambda(n,k)*n + rho(n,k), with lambda(n,k) = (3^X(n,k,-1))/2^k and rho(n,k) = sum(x(n,j)*(3^X(n,k,j))/ 2^(k-j), j=0..(k-1)) with X(n,k,j) = sum(x(n,j+p), p=1.. (k-1-j)) where x(n,j) = T^{(j)}(n) (mod 2). The parity sequence suffices to determine T(n,k).
(End)
Extensions
Name shortened, tabl changed into tabf, Cf. added by Wolfdieter Lang, Mar 20 2014
Comments