A093878 -id:A093878 - cp's OEIS frontend

A089028 a(n) = n+1 where the Hofstadter-Conway Delta A093878(n) >0, otherwise a(n) = 1.

1, 3, 1, 5, 6, 1, 1, 9, 10, 11, 1, 13, 1, 1, 1, 17, 18, 19, 20, 1, 22, 23, 1, 25, 1, 1, 28, 1, 1, 1, 1, 33, 34, 35, 36, 37, 1, 39, 40, 41, 1, 43, 44, 1, 46, 1, 1, 49, 50, 1, 52, 1, 1, 55, 1, 1, 1, 59, 1, 1, 1, 1, 1, 65, 66, 67, 68, 69, 70, 1, 72, 73, 74, 75, 1, 77, 78, 79, 1, 81, 82, 1, 84, 1

Offset: 1

Roger L. Bagula, Nov 12 2003

nonn

Cf. A004001.

Conway[n_Integer?Positive] := Conway[n] =Conway[Conway[n-1]] + Conway[n - Conway[n-1]] Conway[1] = Conway[2] = 1 digits=200 a=Table[If[Conway[n]-Conway[n-1]>0, n, 1], {n, 2, digits}]

A094215 a(n)=A093878(n)-floor(n/phi) where phi=(1+sqrt(5))/2.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 2, 1, 1, 1, 0, 1, 1, 2, 2, 2, 3, 2, 3, 2, 1, 2, 2, 2, 3, 3, 3, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 3, 3, 3, 2, 2, 2, 2, 3, 3, 3, 4, 3, 3, 2, 1, 1, 1, 0, 1, 1, 2, 2, 2, 3, 3, 4, 3, 3, 4, 3, 3, 3, 2, 3

Offset: 1

Benoit Cloitre, May 27 2004

nonn

a(A001519(n))=0

A094217 First differences of A093878.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1

Offset: 1

Benoit Cloitre, May 27 2004

nonn

a(n)=A093878(n+1)-A093878(n)

A095772 Number of times n appears in A093878.

Original entry on oeis.org

2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 5, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 1, 1, 1, 1, 1, 6, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 1, 1, 1, 1, 1, 1, 7, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Benoit Cloitre, Jun 05 2004

nonn

J. Grytczuk, Another variation on Conway's recursive sequence, Discr. Math. 282 (2004), 149-161.

Cf. A004001, A051135.

v=vector(1000,j,1);for(n=3,1000,g=v[v[v[n-1]]]+v[n-v[v[n-1]]];v[n]=g);a(n)=sum(i=1,3*n,if(v[i]-n,0,1))

a(A000045(n)-1) = n-4 for n>4.

A004001 Hofstadter-Conway $10000 sequence: a(n) = a(a(n-1)) + a(n-a(n-1)) with a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 7, 8, 8, 8, 8, 9, 10, 11, 12, 12, 13, 14, 14, 15, 15, 15, 16, 16, 16, 16, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 26, 27, 27, 27, 28, 29, 29, 30, 30, 30, 31, 31, 31, 31, 32, 32, 32, 32, 32, 32, 33, 34, 35, 36, 37, 38, 38, 39, 40, 41, 42

Offset: 1

N. J. A. Sloane

On Jul 15 1988 during a colloquium talk at Bell Labs, John Conway stated that he could prove that a(n)/n -> 1/2 as n approached infinity, but that the proof was extremely difficult. He therefore offered $100 to someone who could find an n_0 such that for all n >= n_0, we have |a(n)/n - 1/2| < 0.05, and he offered $10,000 for the least such n_0. I took notes (a scan of my notebook pages appears below), plus the talk - like all Bell Labs Colloquia at that time - was recorded on video. John said afterwards that he meant to say $1000, but in fact he said $10,000. I was in the front row. The prize was claimed by Colin Mallows, who agreed not to cash the check. - N. J. A. Sloane, Oct 21 2015
a(n) - a(n-1) = 0 or 1 (see the D. Newman reference). - Emeric Deutsch, Jun 06 2005
a(A188163(n)) = n and a(m) < n for m < A188163(n). - Reinhard Zumkeller, Jun 03 2011
From Daniel Forgues, Oct 04 2019: (Start)
Conjectures:
  a(n) = n/2 iff n = 2^k, k >= 1.
  a(n) = 2^(k-1): k times, for n = 2^k - (k-1) to 2^k, k >= 1. (End)

			If n=4, 2^4=16, a(16-i) = 2^(4-1) = 8 for 0 <= i <= 4-1 = 3, hence a(16)=a(15)=a(14)=a(13)=8.

J. Arkin, D. C. Arney, L. S. Dewald and W. E. Ebel, Jr., Families of recursive sequences, J. Rec. Math., 22 (No. 22, 1990), 85-94.
B. W. Conolly, Meta-Fibonacci sequences, in S. Vajda, editor, "Fibonacci and Lucas Numbers and the Golden Section", Halstead Press, NY, 1989, pp. 127-138.
R. K. Guy, Unsolved Problems Number Theory, Sect. E31.
D. R. Hofstadter, personal communication.
C. A. Pickover, Wonders of Numbers, "Cards, Frogs and Fractal sequences", Chapter 96, pp. 217-221, Oxford Univ. Press, NY, 2000.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Vajda, Fibonacci and Lucas Numbers and the Golden Section, Wiley, 1989, see p. 129.
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 129.

T. D. Noe, Table of n, a(n) for n = 1..10000
Altug Alkan, On a Generalization of Hofstadter's Q-Sequence: A Family of Chaotic Generational Structures, Complexity (2018) Article ID 8517125.
Altug Alkan and O. Ozgur Aybar, On Families of Solutions for Meta-Fibonacci Recursions Related to Hofstadter-Conway $10000 Sequence, Slides of talk at presented in 5th International Interdisciplinary Chaos Symposium on Chaos and Complex Systems, May 9-12, 2019.
Altug Alkan, Nathan Fox, and Orhan Ozgur Aybar, On Hofstadter Heart Sequences, Complexity, Volume 2017, Article ID 2614163, 8 pages.
B. Balamohan, A. Kuznetsov and S. Tanny, On the behavior of a variant of Hofstadter's Q-sequence, J. Integer Sequences, Vol. 10 (2007), #07.7.1.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
Nathaniel D. Emerson, A Family of Meta-Fibonacci Sequences Defined by Variable-Order Recursions, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.8.
Christopher S. Flippen, Minimal Sets, Union-Closed Families, and Frankl's Conjecture, Master's thesis, Virginia Commonwealth Univ., 2023.
Nathan Fox, Linear-Recurrent Solutions to Meta-Fibonacci Recurrences, Part 1 (video), Rutgers Experimental Math Seminar, Oct 01 2015. Part 2 is vimeo.com/141111991.
Nathan Fox, A Slow Relative of Hofstadter's Q-Sequence, arXiv:1611.08244 [math.NT], 2016.
S. W. Golomb, Discrete chaos: sequences satisfying "strange" recursions, unpublished manuscript, circa 1990 [cached copy, with permission (annotated)]
J. Grytczuk, Another variation on Conway's recursive sequence, Discr. Math. 282 (2004), 149-161.
R. K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.
R. K. Guy, Letter to N. J. A. Sloane with attachment, 1988
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
Nick Hobson, Python program for this sequence
D. R. Hofstadter, Plot of 100000 terms of a(n)-n/2
D. R. Hofstadter, Analogies and Sequences: Intertwined Patterns of Integers and Patterns of Thought Processes, Lecture in DIMACS Conference on Challenges of Identifying Integer Sequences, Rutgers University, Oct 10 2014; Part 1, Part 2.
A. Isgur, R. Lech, S. Moore, S. Tanny, Y. Verberne, and Y. Zhang, Constructing New Families of Nested Recursions with Slow Solutions, SIAM J. Discrete Math., 30(2), 2016, 1128-1147. (20 pages); DOI:10.1137/15M1040505.
D. Kleitman, Solution to Problem E3274, Amer. Math. Monthly, 98 (1991), 958-959.
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.
C. L. Mallows, Conway's challenge sequence, Amer. Math. Monthly, 98 (1991), 5-20.
D. Newman, Problem E3274, Amer. Math. Monthly, 95 (1988), 555.
John A. Pelesko, Generalizing the Conway-Hofstadter $10,000 Sequence, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.5.
K. Pinn, A chaotic cousin of Conway's recursive sequence, Experimental Mathematics, 9:1 (2000), 55-65.
N. J. A. Sloane, Scan of notebook pages with notes on John Conway's Colloquium talk on Jul 15 1988 [See Comments above]
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
I. Vardi, Email to N. J. A. Sloane, Jun. 1991
Eric Weisstein's World of Mathematics, Hofstadter-Conway 10000-Dollar Sequence.
Eric Weisstein's World of Mathematics, Newman-Conway Sequence
Wikipedia, Hofstadter sequence
Index entries for sequences related to binary expansion of n
Index entries for Hofstadter-type sequences

Cf. A005229, A005185, A080677, A088359, A087686, A093879 (first differences), A265332, A266341, A055748 (a chaotic cousin), A188163 (greedy inverse).
Cf. A004074 (A249071), A005350, A005707, A093878. Different from A086841. Run lengths give A051135.
Cf. also permutations A267111-A267112 and arrays A265901, A265903.

a004001 n = a004001_list !! (n-1)
a004001_list = 1 : 1 : h 3 1  {- memoization -}
  where h n x = x' : h (n + 1) x'
          where x' = a004001 x + a004001 (n - x)
-- Reinhard Zumkeller, Jun 03 2011

[n le 2 select 1 else Self(Self(n-1))+ Self(n-Self(n-1)):n in [1..75]]; // Marius A. Burtea, Aug 16 2019

A004001 := proc(n) option remember; if n<=2 then 1 else procname(procname(n-1)) +procname(n-procname(n-1)); fi; end;

a[1] = 1; a[2] = 1; a[n_] := a[n] = a[a[n - 1]] + a[n - a[n - 1]]; Table[ a[n], {n, 1, 75}] (* Robert G. Wilson v *)

a=vector(100);a[1]=a[2]=1;for(n=3,#a,a[n]=a[a[n-1]]+a[n-a[n-1]]);a \\ Charles R Greathouse IV, Jun 10 2011

first(n)=my(v=vector(n)); v[1]=v[2]=1; for(k=3, n, v[k]=v[v[k-1]]+v[k-v[k-1]]); v \\ Charles R Greathouse IV, Feb 26 2017

def a004001(n):
    A = {1: 1, 2: 1}
    c = 1 #counter
    while n not in A.keys():
        if c not in A.keys():
            A[c] = A[A[c-1]] + A[c-A[c-1]]
        c += 1
    return A[n]
# Edward Minnix III, Nov 02 2015

@CachedFunction
def a(n): # a = A004001
    if n<3: return 1
    else: return a(a(n-1)) + a(n-a(n-1))
[a(n) for n in range(1,101)] # G. C. Greubel, Apr 25 2024

;; An implementation of memoization-macro definec can be found for example from: http://oeis.org/wiki/Memoization
(definec (A004001 n) (if (<= n 2) 1 (+ (A004001 (A004001 (- n 1))) (A004001 (- n (A004001 (- n 1)))))))
;; Antti Karttunen, Oct 22 2014

Limit_{n->infinity} a(n)/n = 1/2 and as special cases, if n > 0, a(2^n-i) = 2^(n-1) for 0 <= i < = n-1; a(2^n+1) = 2^(n-1) + 1. - Benoit Cloitre, Aug 04 2002 [Corrected by Altug Alkan, Apr 03 2017]

A117567 Riordan array ((1+x^2)/(1-x^3),x).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1

Offset: 0

Paul Barry, Mar 29 2006

Sequence array for the sequence F(L((n+2)/3)).

			Triangle begins:
n\k|  0  1  2  3  4  5  6  7  8  9
---+--------------------------------
0  |  1,
1  |  0, 1,
2  |  1, 0, 1,
3  |  1, 1, 0, 1,
4  |  0, 1, 1, 0, 1,
5  |  1, 0, 1, 1, 0, 1,
6  |  1, 1, 0, 1, 1, 0, 1,
7  |  0, 1, 1, 0, 1, 1, 0, 1,
8  |  1, 0, 1, 1, 0, 1, 1, 0, 1,
9  |  1, 1, 0, 1, 1, 0, 1, 1, 0, 1
etc. Row and column numbering added by _Antti Karttunen_, Jan 19 2025

Murat Sahin and Elif Tan, Conditional (strong) divisibility sequences, Fib. Q., 56 (No. 1, 2018), 18-31.

Antti Karttunen, Table of n, a(n) for n = 0..101474; the first 450 rows of the triangle
D. Panario, M. Sahin, Q. Wang, A family of Fibonacci-like conditional sequences, INTEGERS, Vol. 13, 2013, #A78.
Index entries for characteristic functions.

Row sums are A093878. Diagonal sums are A051275. Inverse is A117568.

up_to = 119;
A117567tr0(n,k) = abs(kronecker((n-k+2), 3)); \\ We could also use fibonacci instead of abs
A117567list(up_to) = { my(v = vector(1+up_to), i=0); for(n=0,oo, for(k=0,n, i++; if(i > 1+up_to, return(v)); v[i] = A117567tr0(n,k))); (v); };
v117567 = A117567list(up_to);
A117567(n) = v117567[1+n]; \\ Antti Karttunen, Jan 19 2025

Number triangle T(n,k) = F(L((n-k+2)/3))[k<=n] where L(j/p) is the Legendre symbol of j and p.

In the above, I assume that F stands for Fibonacci sequence (A000045), which in domain {-1, 0, 1} reduces to taking the absolute value of the argument. - Antti Karttunen, Jan 19 2025

Data section extended up to a(119) [15 rows of triangle] by Antti Karttunen, Jan 19 2025

A095769 a(1)=a(2)=1; a(n)=a(a(a(a(n-1))))+a(n-a(a(a(n-1)))).

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 11, 12, 13, 13, 14, 15, 16, 17, 17, 17, 18, 19, 19, 20, 21, 22, 23, 24, 25, 26, 26, 26, 26, 27, 28, 28, 29, 30, 31, 31, 32, 33, 34, 35, 36, 37, 38, 39, 39, 39, 39, 39, 40, 41, 41, 42, 43, 44, 45, 46, 47, 47, 48, 48, 48, 49, 50, 51, 52

Offset: 1

Benoit Cloitre, Jun 05 2004

nonn

A generalization of A004001.

J. Grytczuk, Another variation on Conway's recursive sequence, Discr. Math. 282 (2004), 149-161.

Cf. A004001, A093878, A095770, A095771.

v=vector(150,j,1);for(n=3,150,g=v[v[v[v[n-1]]]]+v[n-v[v[v[n-1]]]];v[n]=g);a(n)=v[n]

For n > 1, a(A000930(n)) = A000930(n-1).

Conjecture: lim_{n->oo} a(n)/n = 0.682327803828... (the real positive root of x^3 + x = 1).

A095770 First differences of A095769.

Original entry on oeis.org

0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0

Offset: 1

Benoit Cloitre, Jun 05 2004

nonn

J. Grytczuk, Another variation on Conway's recursive sequence, Discr. Math. 282 (2004), 149-161.

Cf. A004001, A093878, A095769, A095771.

v=vector(150,j,1);for(n=3,150,g=v[v[v[v[n-1]]]]+v[n-v[v[v[n-1]]]];v[n]=g);a(n)=v[n+1]-v[n]

A095771 Number of times n appears in A095769.

Original entry on oeis.org

2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 5, 1, 2, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Benoit Cloitre, Jun 05 2004

nonn

J. Grytczuk, Another variation on Conway's recursive sequence, Discr. Math. 282 (2004), 149-161.

Cf. A004001, A093878, A095769, A095770, A051135.

v=vector(1000,j,1);for(n=3,1000,g=v[v[v[v[n-1]]]]+v[n-v[v[v[n-1]]]];v[n]=g);a(n)=sum(i=1,3*n,if(v[i]-n,0,1))

a(n) = card{ k in N : A095769(k)=n }.

A320063 A sequence which changes by one or zero: a(n) = a(n-1-a(a(n-1))) + a(a(a(n-1))) for n > 1, a(n) = n for n < 2.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6, 6, 6, 7, 8, 8, 8, 8, 9, 10, 10, 10, 10, 11, 12, 12, 13, 13, 13, 13, 14, 15, 15, 15, 16, 17, 17, 17, 17, 18, 19, 19, 19, 20, 20, 21, 21, 21, 21, 21, 22, 23, 23, 23, 24, 24, 25, 26, 27, 27, 27, 27, 27, 28, 28, 29, 29, 29

Offset: 0

David S. Newman, Oct 04 2018

nonn

This is similar to a problem that I had in the Monthly 35 years ago. The solution then was by Daniel Kleitman.

Alois P. Heinz, Table of n, a(n) for n = 0..65535
Abraham Isgur, Mustazee Rahman, On variants of Conway and Conolly's Meta-Fibonacci recursions, arXiv:1407.0425 [math.CO], 2014.
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.
David Newman and Daniel J. Kleitman, Solution to Problem E3274, Amer. Math. Monthly, 98 (1991), 958-959.

Cf. A093878.

a:= proc(n) option remember; `if`(n<2, n,
      a(n-1-a(a(n-1)))+a(a(a(n-1))))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 08 2018

f[0] = 0;
f[1] = 1;
f[n_] := f[n] = +f[n - 1 - f[f[n - 1]]] + f[f[f[n - 1]]];
Table[f[i], {i, 1, 30}]

cp's OEIS Frontend

A089028 a(n) = n+1 where the Hofstadter-Conway Delta A093878(n) >0, otherwise a(n) = 1.

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A094215 a(n)=A093878(n)-floor(n/phi) where phi=(1+sqrt(5))/2.

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A094217 First differences of A093878.

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A095772 Number of times n appears in A093878.

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A004001 Hofstadter-Conway $10000 sequence: a(n) = a(a(n-1)) + a(n-a(n-1)) with a(1) = a(2) = 1.

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A117567 Riordan array ((1+x^2)/(1-x^3),x).

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A095769 a(1)=a(2)=1; a(n)=a(a(a(a(n-1))))+a(n-a(a(a(n-1)))).

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A095770 First differences of A095769.

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A095771 Number of times n appears in A095769.

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