login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A177789 Irregular triangle read by rows in which row n gives the congruences (mod 2^A020914(n)) satisfied by the numbers having dropping time A122437(n+1) in the Collatz (3x+1) iteration. 5
0, 1, 3, 11, 23, 7, 15, 59, 39, 79, 95, 123, 175, 199, 219, 287, 347, 367, 423, 507, 575, 583, 735, 815, 923, 975, 999, 231, 383, 463, 615, 879, 935, 1019, 1087, 1231, 1435, 1647, 1703, 1787, 1823, 1855, 2031, 2203, 2239, 2351, 2587, 2591, 2907, 2975, 3119 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
The dropping time is the number of Collatz iterations required to reach a lower number than starting value. Garner mentions these congruences. The first term in row n is A122442(n+1) for n > 1. The length of row n is A100982(n). The triangle means:
numbers 0 (mod 2) and > 0 have dropping time 1;
numbers 1 (mod 4) and > 1 have dropping time 3;
numbers 3 (mod 16) have dropping time 6;
numbers 11, 23 (mod 32) have dropping time 8;
numbers 7, 15, 59 (mod 128) have dropping time 11;
numbers 39, 79, 95, 123, 175, 199, 219 (mod 256) have dropping time 13.
Theorem: a(n) can be evaluated using a directed rooted tree produced by a precise algorithm. Each node of this tree is given by a unique Diophantine equation whose only positive solutions are the integers with a finite stopping time. The algorithm generates (in a three step loop) the parity vectors which define the Diophantine equations. The two directions of the construction principle gives the tree a triangular form which extends ever more downwards with each column. There exist explicit arithmetic relationships between the parent and child vertices. As a consequence, a(n) can be generated algorithmically. The algorithm also generates A100982. - Mike Winkler, Sep 12 2017
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..12448 (rows n = 0..13, flattened)
Lynn E. Garner, On the Collatz 3n + 1 Algorithm, Proc. Amer. Math. Soc., Vol. 82(1981), 19-22.
Ruud H.G. van Tol, Perl code
Mike Winkler, New results on the stopping time behaviour of the Collatz 3x + 1 function, arXiv:1504.00212 [math.GM], 2015.
EXAMPLE
Triangle begins:
0;
1;
3;
11, 23;
7, 15, 59;
39, 79, 95, 123, 175, 199, 219;
...
From Mike Winkler, Sep 12 2017: (Start)
The beginning of the directed rooted tree produced by the algorithm of the Theorem. The triangular form can be seen clearly. The way the tree structure is sorting a(n), respectively the residue classes, mirrors the explicit arithmetic relationships mentioned in the Theorem.
3 (mod 2^4) -- 11 (mod 2^5) -- 59 (mod 2^7) -- 123 (mod 2^8) --
| |
| 219 (mod 2^8) --
|
|
23 (mod 2^5) --- 7 (mod 2^7) -- 199 (mod 2^8) --
| |
| 39 (mod 2^8) --
|
|
15 (mod 2^7) --- 79 (mod 2^8) --
|
175 (mod 2^8) --
|
95 (mod 2^8) --
(End)
MATHEMATICA
DroppingTime[n_] := Module[{m=n, k=0}, If[n>1, While[m>=n, k++; If[EvenQ[m], m=m/2, m=3*m+1]]]; k]; dt=Floor[1+Range[0, 20]*Log[2, 6]]; e=Floor[1+Range[0, 20]*Log[2, 3]]; Join[{0, 1}, Flatten[Table[Select[Range[3, 2^e[[n]], 2], DroppingTime[ # ]==dt[[n]] &], {n, 2, 8}]]]
PROG
(PARI) /* algorithm for generating the parity vectors of the Theorem, the tree structure is given by the three STEP's */
{k=3; Log32=log(3)/log(2); limit=14; /*or limit>14*/ T=matrix(limit, 60000); xn=3; /*initial tuple for n=1*/ A=[]; for(i=1, 2, A=concat(A, i)); A[1]=1; A[2]=1; T[1, 1]=A; for(n=2, limit, print("n="n); Sigma=floor(1+(n+1)*Log32); d=floor(n*Log32)-floor((n-1)*Log32); Kappa=floor(n*Log32); Kappa2=floor((n-1)*Log32); r=1; v=1; until(w==0, A=[]; for(i=1, Kappa2+1, A=concat(A, i)); A=T[n-1, v]; B=[]; for(i=1, Kappa+1, B=concat(B, i)); for(i=1, Kappa2+1, B[i]=A[i]); /* STEP 1 */ if(d==1, B[k]=1; T[n, r]=B; r++; v++); if(d==2, B[k]=0; B[k+1]=1; T[n, r]=B; r++; v++); /* STEP 2 */ if(B[Kappa]==0, for(j=1, Kappa-n, B[Kappa+1-j]=B[Kappa+2-j]; B[Kappa+2-j]=0; T[n, r]=B; r++; if(B[Kappa-j]==1, break(1)))); /* STEP 3 */ w=0; for(i=n+2, Kappa+1, w=w+B[i])); k=k+d; p=1; h2=3; for(i=1, r-1, h=0; B=T[n, i]; until(B[h]==0, h++); if(h>h2, p=1; h2++; print); print(T[n, i]" "p" "i); p++); print)} \\ Mike Winkler, Sep 12 2017
CROSSREFS
Cf. A060445 (dropping time of odd numbers), A100982.
Sequence in context: A293766 A178946 A087078 * A289526 A289765 A141226
KEYWORD
nonn,tabf
AUTHOR
T. D. Noe, May 13 2010
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 1 22:05 EDT 2024. Contains 375597 sequences. (Running on oeis4.)