A176014 -id:A176014 - cp's OEIS frontend

A010684 Period 2: repeat (1,3); offset 0.

1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1

Offset: 0

N. J. A. Sloane

Hankel transform is [1,-8,0,0,0,0,0,0,0,0,...]. - Philippe Deléham, Mar 29 2007
Binomial transform gives [1,4,8,16,32,64,...] (A151821(n+1)). - Philippe Deléham, Sep 17 2009
Continued fraction expansion of (3+sqrt(21))/6. - Klaus Brockhaus, May 04 2010
Positive sum of the coordinates from the image of the point (1,-2) after n 90-degree rotations about the origin. - Wesley Ivan Hurt, Jul 06 2013
This sequence can be generated by an infinite number of formulas having the form a^(b*n) mod c where a is congruent to 3 mod 4 and b is any odd number. If a is congruent to 3 mod 4 then c can be 4; if a is also congruent to 3 mod 8 then c can be 8. For example: a(n)= 15^(3*n) mod 4, a(n) = 19^(5*n) mod 4, a(n) = 19^(5*n) mod 8. - Gary Detlefs, May 19 2014
This sequence is also the unsigned periodic Schick sequence for p = 5. See the Schick reference, p. 158, for p = 5.- Wolfdieter Lang, Apr 03 2020
Digits following the decimal point when 1/3 is converted to base 5. - Jamie Robert Creasey, Oct 15 2021
Decimal expansion of 13/99. - Stefano Spezia, Feb 09 2025

			0.131313131313131313131313131313131313131313131...

Carl Schick, Trigonometrie und unterhaltsame Zahlentheorie, Bokos Druck, Zürich, 2003 (ISBN 3-9522917-0-6). Tables 3.1 to 3.10, for odd p = 3..113 (with gaps), pp. 158-166.

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A112030, A112033, A176014 (decimal expansion of (3+sqrt(21))/6).

[seq (modp((2*n+1),4),n=0..80)]; # Zerinvary Lajos, Nov 30 2006

Table[2-(-1)^n, {n, 0, 100}] (* Wesley Ivan Hurt, Mar 24 2014 *)
PadRight[{},120,{1,3}] (* Harvey P. Dale, Mar 11 2025 *)

a(n)=1+n%2*2 \\ Charles R Greathouse IV, Dec 28 2011

def A010684(n): return 3 if n&1 else 1 # Chai Wah Wu, Jan 17 2023

[power_mod(3, n, 8)for n in range(0, 81)] # Zerinvary Lajos, Nov 24 2009

From Paul Barry, Apr 29 2003: (Start)
a(n) = 2-(-1)^n.
G.f.: (1+3x)/((1-x)(1+x)).
E.g.f.: 2*exp(x) - exp(-x). (End)
a(n) = 2*A153643(n) - A153643(n+1). - Paul Curtz, Dec 30 2008
a(n) = 3^(n mod 2). - Jaume Oliver Lafont, Mar 27 2009
a(n) = 7^n mod 4. - Vincenzo Librandi, Feb 07 2011
a(n) = 1 + 2*(n mod 2). - Wesley Ivan Hurt, Jul 06 2013
a(n) = A000034(n) + A000035(n). - James Spahlinger, Feb 14 2016

A005186 a(n) is the number of integers m which take n steps to reach 1 in '3x+1' problem.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 4, 4, 6, 6, 8, 10, 14, 18, 24, 29, 36, 44, 58, 72, 91, 113, 143, 179, 227, 287, 366, 460, 578, 732, 926, 1174, 1489, 1879, 2365, 2988, 3780, 4788, 6049, 7628, 9635, 12190, 15409, 19452, 24561, 31025, 39229, 49580, 62680, 79255, 100144

Offset: 0

N. J. A. Sloane, R. K. Guy

Appears to settle into approximately exponential growth after about 25 terms or so with a ratio between adjacent terms of roughly 1.264. - Howard A. Landman, May 24 2003
David W. Wilson (Jun 10 2003) gives a heuristic argument that the constant should be the largest eigenvalue of the matrix [ 1 0 0 1 0 0 / 0 0 0 0 1/3 0 / 0 1 0 0 1 0 / 0 0 0 0 1/3 0 / 0 0 1 0 0 1 / 0 0 0 0 1/3 0 ], which is (3 + sqrt(21))/6 = 1.2637626... = A176014.
Merlini and Sala (1999) deduce the value (1 + sqrt(7/3))/2 (A176014) for the asymptotic ratio a(n+1)/a(n) for n -> oo and call this "Collatz's constant". This is the same value as the constant mentioned above, see the "Heuristic argument for the asymptotic value (3 + sqrt(21))/6 of the ratio a(n+1)/a(n)" note in the Links section. - Markus Sigg, Nov 27 2020

R. K. Guy, personal communication.
J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010; see p. 33.
Danilo Merlini and Nicoletta Sala, On the Fibonacci's Attractor and the Long Orbits in the 3n+1 Problem, International Journal of Chaos Theory and Applications, Vol. 4, No. 2-3 (1999), 75-84.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Markus Sigg, Table of n, a(n) for n = 0..125 (first 71 terms from T. D. Noe, terms up to n = 120 from Bert Dobbelaere).
S. N. Anderson, Struggling with the 3x+1 problem, Math. Gazette, 71 (1987), 271-274.
S. N. Anderson, Struggling with the 3x+1 problem, Math. Gazette, 71 (1987), 271-274. [Annotated scanned copy]
R. K. Guy, S. N. Anderson, and N. J. A. Sloane, Correspondence, 1988.
Jeffrey C. Lagarias, The 3x+1 Problem: An Overview, arXiv:2111.02635 [math.NT], 2021.
Wolfdieter Lang, On Collatz Words, Sequences, and Trees, Journal of Integer Sequences, Vol 17 (2014), Article 14.11.7.
Hugo Pfoertner, Ratio of successive terms, illustration of deviation from (3+sqrt(21))/6.
Markus Sigg, Heuristic argument for the asymptotic value (3+sqrt(21))/6 of the ratio a(n+1)/a(n).
Wikipedia, The beginning of the Collatz directed graph
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A088975, A176014, A176866.

(* This program is not suitable to compute more than 20 terms *) maxiSteps = 20; mMaxi = 2*10^6; Clear[a]; a[] = 0; steps[m] := steps[m] = Module[{n = m, ns = 0}, While[n != 1, If[Mod[n, 2] == 0, n = n/2, n = 3*n+1]; ns++]; ns]; Do[sn = steps[m]; If[sn <= maxiSteps, a[sn] = a[sn]+1; Print["m = ", m, " a(", sn, ") = ", a[sn]]], {m, 1, mMaxi}]; Table[a[n], {n, 0, maxiSteps}] (* Jean-François Alcover, Oct 18 2013 *)
(* 60 terms in under a minute *) s = {1}; t = Join[{1}, Table[s = Union[2*s, (Select[s, Mod[#, 3] == 1 && OddQ[(# - 1)/3] && (# - 1)/3 > 1 &] - 1)/3]; Length[s], {n, 60}]] (* T. D. Noe, Oct 18 2013 *)

first(n)=my(v=vector(n+1), u=[1], old=u, w); v[1]=1; for(i=1, n, w=List(); for(j=1, #u, listput(w, 2*u[j]); if(u[j]%6==4, listput(w, u[j]\3))); old=setunion(old, u); u=setminus(Set(w), old); v[i+1]=#u); v \\ Charles R Greathouse IV, Jun 26 2017

first(n)={my(v=Vec([1, 1, 1, 1, 1], n+1), u=[16]); for(i=5, n, u=concat(2*u, [x\3 | x<-u, x%6==4 ]); v[i+1]=#u); v} \\ Joe Slater, Sep 01 2024

#!/usr/bin/perl @old = ( 1 ); while (1) { print scalar(@old), " "; @new = ( ); foreach $n (@old) { $used{$n} = 1; if (($n % 6) == 4) { $m = ($n-1)/3; push(@new,$m) unless ($used{$m}); } $m = $n + $n; push(@new,$m) unless ($used{$m}); } @old = @new; }

def search(x,d,lst):
    while d>0:
        lst[d]+=1
        if x%6==4 and x>4:
            search(x//3,d-1,lst)
        x*=2
        d-=1
    lst[d]+=1
def A005186_list(n_max):
    lst=[0]*(n_max+1)
    search(1,n_max,lst)
    return lst[::-1]
# Bert Dobbelaere, Sep 09 2018

More terms from Larry Reeves (larryr(AT)acm.org), Apr 27 2001

A090458 Decimal expansion of (3 + sqrt(21))/2.

Original entry on oeis.org

3, 7, 9, 1, 2, 8, 7, 8, 4, 7, 4, 7, 7, 9, 2, 0, 0, 0, 3, 2, 9, 4, 0, 2, 3, 5, 9, 6, 8, 6, 4, 0, 0, 4, 2, 4, 4, 4, 9, 2, 2, 2, 8, 2, 8, 8, 3, 8, 3, 9, 8, 5, 9, 5, 1, 3, 0, 3, 6, 2, 1, 0, 6, 1, 9, 5, 3, 4, 3, 4, 2, 1, 2, 7, 7, 3, 8, 8, 5, 4, 4, 3, 3, 0, 2, 1, 8, 0, 7, 7, 9, 7, 4, 6, 7, 2, 2, 5, 1, 6, 3

Offset: 1

Felix Tubiana, Feb 05 2004

Decimal expansion of the solution to n/x = x-n for n-3. n/x = x-n with n=1 gives the Golden Ratio = 1.6180339887...
n/x = x-n ==> x^2 - n*x - n = 0 ==> x = (n + sqrt(n^2 + 4*n)) / 2 (Positive Root) n = 3: x = (3 + sqrt(21))/2 = 3.79128784747792...
x=3.7912878474... is the shape of a rectangle whose geometric partition (as at A188635) consists of 3 squares, then 1 square, then 3 squares, etc., matching the continued fraction of x, which is [3,1,3,1,3,1,3,1,3,1,...]. (See the Mathematica program below.) - Clark Kimberling, May 05 2011
x appears to be the limit for n to infinity of the ratio of the number of even numbers that take n steps to reach 1 to the number of odd numbers that take n steps to reach 1 in the Collatz iteration. As A005186(n-1) is the number of even numbers that take n steps to reach 1, this means x = lim A005186(n-1)/A176866(n). - Markus Sigg, Oct 20 2020
From Wolfdieter Lang, Sep 02 2022: (Start)
This integer in the quadratic number field Q(sqrt(21)) equals the (real) cube root of 27 + 6*sqrt(21) = 54.4954541... . See Euler, Elements of Algebra, Article 748 or Algebra (in German) p. 306, Kapitel 12, 187.
Subtracting 3 from the present number gives the (real) cube root of
  -27 +  6*sqrt(21) = 0.4954541... . (End)

			3.79128784747792...

Leonhard Euler, Vollständige Anleitung zur Algebra, (1770), Reclam, Leipzig, 1883, p.306, Kapitel 12, 187.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Leonhard Euler, Elements of Algebra, p. 244, Article 748.
Markus Sigg, Heuristic argument for the asymptotic value of the even/odd ratio of number of steps in the Collatz iteration, 2020.
Index entries for algebraic numbers, degree 2

Of the same type as this: A090388 (n=2), A090488 (n=4), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090654 (n=8), A090655 (n=9), A090656 (n=10).
Equals 3*A176014 (constant).
Cf. A356034.

FromContinuedFraction[{3,1,{3,1}}]
ContinuedFraction[%,20]
RealDigits[N[%%,120]] (*A090458*)
N[%%%,40]
RealDigits[(3 + Sqrt[21])/2, 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

solve(x=3,4,x^2-3*x-3) \\ Charles R Greathouse IV, Oct 04 2011

(3+sqrt(21))/2 \\ Charles R Greathouse IV, Oct 04 2011

Equals (27 + 6*sqrt(21))^(1/3). - Wolfdieter Lang, Sep 01 2022

Additional comments from Rick L. Shepherd, Jul 02 2004

A176866 The number of odd numbers that require n Collatz (3x+1) iterations to reach 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 2, 4, 4, 6, 5, 7, 8, 14, 14, 19, 22, 30, 36, 48, 60, 79, 94, 118, 154, 194, 248, 315, 390, 486, 623, 792, 1008, 1261, 1579, 2007, 2555, 3219, 4043, 5109, 6464, 8204, 10351, 13100, 16575, 20889, 26398, 33388, 42155, 53370, 67414

Offset: 0

T. D. Noe, Apr 27 2010

nonn

Both the 3x+1 steps and the halving steps are counted. The asymptotic growth rate appears to be the same as A005186, about 1.26 (A176014).
a(n) is, for n >= 4, the number of 4 (mod 6) nodes (vertices) of row n-1 of the Collatz tree A127824. The node 4 has in A127824 outdegree 1 in order to avoid a repetition of the whole tree. - Wolfdieter Lang, Mar 26 2014
The heuristic arguments given in the LINKS of A005186 suggest that this sequence has the same asymptotic growth rate (3+sqrt(21))/6. - Markus Sigg, Sep 07 2024

			23, 141, 151, 853, 909, and 5461 are the only odd numbers that require exactly 15 iterations to reach 1. Hence a(15)=6.
At row 15 with a(16) = 5 nodes 4 (mod 6) the left-right symmetry for the number of 4 (mod 6) nodes in the Collatz tree A127824 is broken for the first time: in the left half of the tree there are the three nodes 22, 136 and 832 but on the right half only the two nodes 904 and 5440. - _Wolfdieter Lang_, Mar 26 2014

Markus Sigg, Table of n, a(n) for n = 0..125 (first 71 terms from T. D. Noe).
Wolfdieter Lang, On Collatz' Words, Sequences and Trees, arXiv preprint arXiv:1404.2710, 2014 and J. Int. Seq. 17 (2014) # 14.11.7.

Cf. A005186 (number of numbers having stopping time n).
Cf. A127824 (numbers having stopping time n).
Cf. A176014, A176867, A176868.

A182229 a(n) = a(n-1) + floor(a(n-2)/3) with a(0)=2, a(1)=3.

Original entry on oeis.org

2, 3, 3, 4, 5, 6, 7, 9, 11, 14, 17, 21, 26, 33, 41, 52, 65, 82, 103, 130, 164, 207, 261, 330, 417, 527, 666, 841, 1063, 1343, 1697, 2144, 2709, 3423, 4326, 5467, 6909, 8731, 11034, 13944, 17622, 22270, 28144, 35567, 44948, 56803, 71785, 90719, 114647, 144886, 183101

Offset: 0

Alex Ratushnyak, Apr 19 2012

nonn

a(n)/a(n-1) tends to (3+sqrt(21))/6 = 1.263762615825973334... [Bruno Berselli, Apr 23 2012]

Bruno Berselli, Table of n, a(n) for n = 0..1000

Cf. A064324, A182281, A182230; A176014.

a182229 n = a182229_list !! n
a182229_list = 2 : 3 : zipWith (+)
                       (map (flip div 3) a182229_list) (tail a182229_list)
-- Reinhard Zumkeller, Apr 30 2015

[n le 2 select n+1 else Self(n-1)+Floor(Self(n-2)/3): n in [1..51]]; // Bruno Berselli, Apr 21 2012

RecurrenceTable[{a[0] == 2, a[1] == 3, a[n] == a[n - 1] + Floor[a[n - 2]/3]}, a, {n, 50}] (* Bruno Berselli, Apr 21 2012 *)

prpr = 2
prev = 3
for i in range(2,51):
    current = prev + prpr//3
    print(current, end=',')
    prpr = prev
    prev = current

cp's OEIS Frontend

A010684 Period 2: repeat (1,3); offset 0.

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A005186 a(n) is the number of integers m which take n steps to reach 1 in '3x+1' problem.

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A090458 Decimal expansion of (3 + sqrt(21))/2.

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Mathematica

PARI

PARI

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Extensions

A176866 The number of odd numbers that require n Collatz (3x+1) iterations to reach 1.

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Author

Keywords

Comments

Examples

Links

Crossrefs

A182229 a(n) = a(n-1) + floor(a(n-2)/3) with a(0)=2, a(1)=3.

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Programs

Haskell

Magma

Mathematica

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