A051135 -id:A051135 - cp's OEIS frontend

A267110 If A051135(n) = 1, then a(n) = A004001(n) - 1, otherwise a(n) = n - A004001(n).

0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 5, 7, 6, 7, 8, 8, 9, 10, 11, 9, 12, 13, 10, 14, 11, 12, 15, 13, 14, 15, 16, 16, 17, 18, 19, 20, 17, 21, 22, 23, 18, 24, 25, 19, 26, 20, 21, 27, 28, 22, 29, 23, 24, 30, 25, 26, 27, 31, 28, 29, 30, 31, 32, 32, 33, 34, 35, 36, 37, 33, 38, 39, 40, 41, 34, 42, 43, 44, 35, 45, 46, 36, 47

Offset: 1

Antti Karttunen, Jan 16 2016

For n > 1, a(n) gives the contents of the parent of the node which contains n in A267112-tree.
Each n > 0 occurs exactly twice, in positions A088359(n) and A087686(n+1).
The sequence maps each n > 1 to a number which is one digit shorter in binary system (cf. "Other identities"). This follows because A004001 is monotonic and A004001(2^n) = 2^(n-1) (see properties (2) and (3) given on page 227 of Kubo & Vakil paper, or page 3 in PDF), and also how the frequency counts Q_n for A004001 are recursively constructed (see Kubo & Vakil paper, p. 229 or A265332 for the illustration).

Antti Karttunen, Table of n, a(n) for n = 1..8192
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.
Index entries for Hofstadter-type sequences

Cf. A004001, A051135, A070939, A087686, A088359, A265332, A267108, A267109, A267112.

(define (A267110 n) (if (= 1 (A051135 n)) (- (A004001 n) 1) (- n (A004001 n))))

If A051135(n) = 1 [Equally: if A265332(n) = 1], then a(n) = A004001(n) - 1, otherwise a(n) = n - A004001(n).
Other identities. For all n >= 2:
A070939(a(n)) = A070939(n) - 1. [See Comments section.]

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]

A087686 Elements of A004001 that repeat consecutively.

Original entry on oeis.org

1, 2, 4, 7, 8, 12, 14, 15, 16, 21, 24, 26, 27, 29, 30, 31, 32, 38, 42, 45, 47, 48, 51, 53, 54, 56, 57, 58, 60, 61, 62, 63, 64, 71, 76, 80, 83, 85, 86, 90, 93, 95, 96, 99, 101, 102, 104, 105, 106, 109, 111, 112, 114, 115, 116, 118, 119, 120, 121, 123, 124, 125, 126, 127

Offset: 1

Roger L. Bagula, Sep 27 2003

nonn

Complement of A088359; A051135(a(n)) > 1. [Reinhard Zumkeller, Jun 03 2011]
From Antti Karttunen, Jan 18 2016: (Start)
This set of numbers is closed with respect to A004001, see A266188.
After 1, one more than the positions of zeros in A093879.
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A004001, A051135, A093879.
Cf. A088359 (complement), A188163 (almost complement).
Cf. A080677 (the least monotonic left inverse).
Cf. also A266188, A265901, A265903, A267110, A267111, A267112.

import Data.List (findIndices)
a087686 n = a087686_list !! (n-1)
a087686_list = map succ $ findIndices (> 1) a051135_list
-- Reinhard Zumkeller, Jun 03 2011
(Scheme, with Antti Karttunen's IntSeq-library)
(define A087686 (MATCHING-POS 1 1 (lambda (n) (> (A051135 n) 1))))
;; Antti Karttunen, Jan 18 2016

Conway[n_Integer?Positive] := Conway[n] =Conway[Conway[n-1]] + Conway[n - Conway[n-1]] Conway[1] = Conway[2] = 1 digits=1000 a=Table[Conway[n], {n, 1, digits}]; b=Table[If[a[[n]]-a[[n-1]]==0, a[[n]], 0], {n, 2, digits}]; c=Delete[Union[b], 1]

Other identities. For all n >= 1:

A080677(a(n)) = n. [See comments in A080677.]

A088359 Numbers which occur only once in A004001.

Original entry on oeis.org

3, 5, 6, 9, 10, 11, 13, 17, 18, 19, 20, 22, 23, 25, 28, 33, 34, 35, 36, 37, 39, 40, 41, 43, 44, 46, 49, 50, 52, 55, 59, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 77, 78, 79, 81, 82, 84, 87, 88, 89, 91, 92, 94, 97, 98, 100, 103, 107, 108, 110, 113, 117, 122, 129, 130, 131, 132

Offset: 1

Robert G. Wilson v, Sep 26 2003

nonn

Out of the first one million terms (a(10^6) = 510403), 258661 occur only once.
Complement of A087686; A051135(a(n)) = 1. - Reinhard Zumkeller, Jun 03 2011
From Antti Karttunen, Jan 18 2016: (Start)
In general, out of the first 2^(n+1) terms of A004001, 2^(n-1) - 1 terms (a quarter) occur only once. See also illustration in A265332.
One more than the positions of ones in A093879.
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Positions of ones in A051135.
Cf. A188163 (same sequence with prepended 1).
Cf. A087686 (complement).
Cf. A004001, A093879, A265332, A266399.
Cf. also A267110, A267111, A267112.

import Data.List (elemIndices)
a088359 n = a088359_list !! (n-1)
a088359_list = map succ $ elemIndices 1 a051135_list
-- Reinhard Zumkeller, Jun 03 2011
(Scheme, with Antti Karttunen's IntSeq-library)
(define A088359 (ZERO-POS 1 1 (COMPOSE -1+ A051135)))
;; Antti Karttunen, Jan 18 2016

a[1] = 1; a[2] = 1; a[n_] := a[n] = a[ a[n - 1]] + a[n - a[n - 1]]; hc = Table[ a[n], {n, 1, 261}]; RunLengthEncodeOne[x_List] := Length[ # ] == 1 & /@ Split[x]; r = RunLengthEncodeOne[hc]; Select[ Range[ Length[r]], r[[ # ]] == True &]

From Antti Karttunen, Jan 18 2016: (Start)
Other identities.
For all n >= 0, a(A000079(n)) = A000051(n+1), that is, a(2^n) = 2^(n+1) + 1.
For all n >= 1:
a(n) = A004001(A266399(n)).
(End)

A265903 Square array read by descending antidiagonals: A(1,k) = A188163(k), and for n > 1, A(n,k) = A087686(1+A(n-1,k)).

Original entry on oeis.org

1, 3, 2, 5, 7, 4, 6, 12, 15, 8, 9, 14, 27, 31, 16, 10, 21, 30, 58, 63, 32, 11, 24, 48, 62, 121, 127, 64, 13, 26, 54, 106, 126, 248, 255, 128, 17, 29, 57, 116, 227, 254, 503, 511, 256, 18, 38, 61, 120, 242, 475, 510, 1014, 1023, 512, 19, 42, 86, 125, 247, 496, 978, 1022, 2037, 2047, 1024, 20, 45, 96, 192, 253, 502, 1006, 1992, 2046, 4084, 4095, 2048

Offset: 1

Antti Karttunen, Dec 18 2015

Square array A(n,k) [where n is row and k is column] is read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

For n >= 3, each row n lists the numbers that appear n times in A004001. See also A051135.

			The top left corner of the array:
     1,    3,    5,    6,     9,    10,    11,    13,    17,    18,    19
     2,    7,   12,   14,    21,    24,    26,    29,    38,    42,    45
     4,   15,   27,   30,    48,    54,    57,    61,    86,    96,   102
     8,   31,   58,   62,   106,   116,   120,   125,   192,   212,   222
    16,   63,  121,  126,   227,   242,   247,   253,   419,   454,   469
    32,  127,  248,  254,   475,   496,   502,   509,   894,   950,   971
    64,  255,  503,  510,   978,  1006,  1013,  1021,  1872,  1956,  1984
   128,  511, 1014, 1022,  1992,  2028,  2036,  2045,  3864,  3984,  4020
   256, 1023, 2037, 2046,  4029,  4074,  4083,  4093,  7893,  8058,  8103
   512, 2047, 4084, 4094,  8113,  8168,  8178,  8189, 16006, 16226, 16281
  1024, 4095, 8179, 8190, 16292, 16358, 16369, 16381, 32298, 32584, 32650
  ...

Inverse permutation: A267104.
Transpose: A265901.
Row 1: A188163.
Row 2: A266109.
Row 3: A267103.
For the known and suspected columns, see the rows listed for transposed array A265901.
Cf. A004001, A051135, A087686.
Cf. A265900 (main diagonal), A265909 (submain diagonal).
Cf. A162598 (column index of n in array), A265332 (row index of n in array).
Cf. also permutations A267111, A267112.

(define (A265903 n) (A265903bi (A002260 n) (A004736 n)))
(define (A265903bi row col) (if (= 1 row) (A188163 col) (A087686 (+ 1 (A265903bi (- row 1) col)))))

For the first row n=1, A(1,k) = A188163(k), for rows n > 1, A(n,k) = A087686(1+A(n-1,k)).

A093879 First differences of A004001.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, May 27 2004

All the terms are 0 or 1: it is easy to show that if {b(n)} = A004001, b(n)>=b(n-1) and b(n)Benoit Cloitre, Jun 05 2004

R. J. Mathar, Table of n, a(n) for n = 1..9999
J. Grytczuk, Another variation on Conway's recursive sequence, Discr. Math. 282 (2004), 149-161.
D. Newman, Problem E3274, Amer. Math. Monthly, 95 (1988), 555.

Cf. A004001, A051135, A087686, A088359, A188163.

h:=[n le 2 select 1 else Self(Self(n-1)) + Self(n-Self(n-1)): n in [1..160]];
A093879:= func< n | h[n+1] - h[n] >;
[A093879(n): n in [1..120]]; // G. C. Greubel, May 19 2024

a[1] = a[2] = 1; a[n_] := a[n] = a[a[n - 1]] + a[n - a[n - 1]]; t = Table[a[n], {n, 110}]; Drop[t, 1] - Drop[t, -1] (* Robert G. Wilson v, May 28 2004 *)

{m=106;v=vector(m,j,1);for(n=3,m,a=v[v[n-1]]+v[n-v[n-1]];v[n]=a);for(n=2,m,print1(v[n]-v[n-1],","))}

@CachedFunction
def h(n): return 1 if (n<3) else h(h(n-1)) + h(n - h(n-1))
def A093879(n): return h(n+1) - h(n)
[A093879(n) for n in range(1,101)] # G. C. Greubel, May 19 2024

(define (A093879 n) (- (A004001 (+ 1 n)) (A004001 n))) ;; Code for A004001 given in that entry. - Antti Karttunen, Jan 18 2016

From Antti Karttunen, Jan 18 2016: (Start)
a(n) = A004001(n+1) - A004001(n).
Other identities. For all n >= 1:
a(A087686(n+1)-1) = 0.
a(A088359(n)-1) = 1.
a(n) = 1 if and only if A051135(n+1) = 1.
(End)

More terms and PARI code from Klaus Brockhaus and Robert G. Wilson v, May 27 2004

A188163 Smallest m such that A004001(m) = n.

Original entry on oeis.org

1, 3, 5, 6, 9, 10, 11, 13, 17, 18, 19, 20, 22, 23, 25, 28, 33, 34, 35, 36, 37, 39, 40, 41, 43, 44, 46, 49, 50, 52, 55, 59, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 77, 78, 79, 81, 82, 84, 87, 88, 89, 91, 92, 94, 97, 98, 100, 103, 107, 108, 110, 113, 117, 122

Offset: 1

Reinhard Zumkeller, Jun 03 2011

nonn

How is this related to A088359? - R. J. Mathar, Jan 09 2013
It is not hard to show that a(n) exists for all n, and in particular a(n) < 2^n. - Charles R Greathouse IV, Jan 13 2013
From Antti Karttunen, Jan 10 & 18 2016: (Start)
Positions of records in A004001. After 1 the positions where A004001 increases (by necessity by one).
An answer to the question of R. J. Mathar above: This sequence is equal to A088359 with prepended 1. This follows because at each of its unique values (terms of A088359), A004001 must grow, but it can grow nowhere else. See Kubo and Vakil paper and especially the illustrations of Q and R-trees on pages 229-230 (pages 5 & 6 in PDF) and also in sequence A265332.
Obviously A004001 can obtain unique values only at points which form a subset (A266399) of this sequence.
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.
User "Panurge", Frankl's conjecture and Oeis sequence A188163, Mathoverflow.net, Mar 29 2016.
Eric Weisstein's World of Mathematics, Hofstadter-Conway $10,000 Sequence.
Wikipedia, Hofstadter sequence
Index entries for Hofstadter-type sequences

Equal to A088359 with prepended 1.
Column 1 of A265901, Row 1 of A265903.
Cf. A051135 (first differences).
Cf. A087686 (complement, apart from the initial 1).
Cf. A004001 (also the least monotonic left inverse of this sequence).
Cf. A093879, A265332.
Cf. A266399 (a subsequence).

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a188163 n = succ $ fromJust $ elemIndex n a004001_list

h:=[n le 2 select 1 else Self(Self(n-1)) + Self(n - Self(n-1)): n in [1..500]]; // h=A004001
A188163:= function(n)
   for j in [1..2*n+1] do
       if h[j] eq n then return j; end if;
   end for;
end function;
[A188163(n): n in [1..100]]; // G. C. Greubel, May 20 2024

A188163 := proc(n)
    for a from 1 do
        if A004001(a) = n then
            return a;
        end if;
    end do:
end proc: # R. J. Mathar, May 15 2013

h[1] = 1; h[2] = 1; h[n_] := h[n] = h[h[n-1]] + h[n - h[n-1]];
a[n_] := For[m = 1, True, m++, If[h[m] == n, Return[m]]];
Array[a, 64] (* Jean-François Alcover, Jan 27 2018 *)

@CachedFunction
def h(n): return 1 if (n<3) else h(h(n-1)) + h(n - h(n-1)) # h=A004001
def A188163(n):
    for j in range(1,2*n+2):
        if h(j)==n: return j
[A188163(n) for n in range(1,101)] # G. C. Greubel, May 20 2024

(define A188163 (RECORD-POS 1 1 A004001))
;; Code for A004001 given in that entry. Uses also my IntSeq-library. - Antti Karttunen, Jan 18 2016

Other identities. For all n >= 1:
A004001(a(n)) = n and A004001(m) < n for m < a(n).
A051135(n) = a(n+1) - a(n).

A265332 a(n) is the index of the column in A265901 where n appears; also the index of the row in A265903 where n appears.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 09 2016

nonn

If all 1's are deleted, the remaining terms are the sequence incremented. - after Franklin T. Adams-Watters Oct 05 2006 comment in A051135.

Ordinal transform of A162598.

			Illustration how the sequence can be constructed by concatenating the frequency counts Q_n of each successive level n of A004001-tree:
--
             1                                      Q_0 = (1)
             |
            _2__                                    Q_1 = (2)
           /    \
         _3    __4_____                             Q_2 = (1,3)
        /     /  |     \
      _5    _6  _7    __8___________                Q_3 = (1,1,2,4)
     /     /   / |   /  |  \        \
   _9    10  11 12  13  14  15___    16_________    Q_4 = (1,1,1,2,1,2,3,5)
  /     /   /  / |  /  / |   |\  \   | \  \  \  \
17    18  19 20 21 22 23 24 25 26 27 28 29 30 31 32
--
The above illustration copied from the page 229 of Kubo and Vakil paper (page 5 in PDF).

Antti Karttunen, Table of n, a(n) for n = 1..8192
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.

Essentially same as A051135 apart from the initial term, which here is set as a(1)=1.
Cf. A004001, A265901, A265903.
Cf. A162598 (corresponding other index).
Cf. A265754.
Cf. also A267108, A267109, A267110.

terms = 120;
h[1] = 1; h[2] = 1;
h[n_] := h[n] = h[h[n - 1]] + h[n - h[n - 1]];
seq[nmax_] := seq[nmax] = (Length /@ Split[Sort @ Array[h, nmax, 2]])[[;; terms]];
seq[nmax = 2 terms];
seq[nmax += terms];
While[seq[nmax] != seq[nmax - terms], nmax += terms];
seq[nmax] (* Jean-François Alcover, Dec 19 2021 *)

(define (A265332 n) (if (= 1 n) 1 (A051135 n)))

a(1) = 1; for n > 1, a(n) = A051135(n).

A265901 Square array read by descending antidiagonals: A(n,1) = A188163(n), and for k > 1, A(n,k) = A087686(1+A(n,k-1)).

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 8, 15, 12, 6, 16, 31, 27, 14, 9, 32, 63, 58, 30, 21, 10, 64, 127, 121, 62, 48, 24, 11, 128, 255, 248, 126, 106, 54, 26, 13, 256, 511, 503, 254, 227, 116, 57, 29, 17, 512, 1023, 1014, 510, 475, 242, 120, 61, 38, 18, 1024, 2047, 2037, 1022, 978, 496, 247, 125, 86, 42, 19, 2048, 4095, 4084, 2046, 1992, 1006, 502, 253, 192, 96, 45, 20

Offset: 1

Antti Karttunen, Dec 18 2015

Square array read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
The topmost row (row 1) of the array is A000079 (powers of 2), and in general each row 2^k contains the sequence (2^n - k), starting from the term (2^(k+1) - k). This follows from the properties (3) and (4) of A004001 given on page 227 of Kubo & Vakil paper (page 3 in PDF).
Moreover, each row 2^k - 1 (for k >= 2) contains the sequence 2^n - n - (k-2), starting from the term (2^(k+1) - (2k-1)). To see why this holds, consider the definitions of sequences A162598 and A265332, the latter which also illustrates how the frequency counts Q_n for A004001 are recursively constructed (in the Kubo & Vakil paper).

			The top left corner of the array:
   1,  2,   4,   8,  16,   32,   64,  128,  256,   512,  1024, ...
   3,  7,  15,  31,  63,  127,  255,  511, 1023,  2047,  4095, ...
   5, 12,  27,  58, 121,  248,  503, 1014, 2037,  4084,  8179, ...
   6, 14,  30,  62, 126,  254,  510, 1022, 2046,  4094,  8190, ...
   9, 21,  48, 106, 227,  475,  978, 1992, 4029,  8113, 16292, ...
  10, 24,  54, 116, 242,  496, 1006, 2028, 4074,  8168, 16358, ...
  11, 26,  57, 120, 247,  502, 1013, 2036, 4083,  8178, 16369, ...
  13, 29,  61, 125, 253,  509, 1021, 2045, 4093,  8189, 16381, ...
  17, 38,  86, 192, 419,  894, 1872, 3864, 7893, 16006, 32298, ...
  18, 42,  96, 212, 454,  950, 1956, 3984, 8058, 16226, 32584, ...
  19, 45, 102, 222, 469,  971, 1984, 4020, 8103, 16281, 32650, ...
  20, 47, 105, 226, 474,  977, 1991, 4028, 8112, 16291, 32661, ...
  22, 51, 112, 237, 490,  999, 2020, 4065, 8158, 16347, 32728, ...
  23, 53, 115, 241, 495, 1005, 2027, 4073, 8167, 16357, 32739, ...
  25, 56, 119, 246, 501, 1012, 2035, 4082, 8177, 16368, 32751, ...
  28, 60, 124, 252, 508, 1020, 2044, 4092, 8188, 16380, 32764, ...
  ...

Antti Karttunen, Table of n, a(n) for n = 1..210; the first 20 antidiagonals of array
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.
Index entries for Hofstadter-type sequences
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A267102.
Transpose: A265903.
Cf. A265900 (main diagonal).
Cf. A162598 (row index of n in array), A265332 (column index of n in array).
Cf. A004001, A051135, A088359, A087686.
Column 1: A188163.
Column 2: A266109.
Row 1: A000079 (2^n).
Row 2: A000225 (2^n - 1, from 3 onward).
Row 3: A000325 (2^n - n, from 5 onward).
Row 4: A000918 (2^n - 2, from 6 onward).
Row 5: A084634 (?, from 9 onward).
Row 6: A132732 (2^n - 2n + 2, from 10 onward).
Row 7: A000295 (2^n - n - 1, from 11 onward).
Row 8: A036563 (2^n - 3).
Row 9: A084635 (?, from 17 onward).
Row 12: A048492 (?, from 20 onward).
Row 13: A249453 (?, from 22 onward).
Row 14: A183155 (2^n - 2n + 1, from 23 onward. Cf. also A244331).
Row 15: A000247 (2^n - n - 2, from 25 onward).
Row 16: A028399 (2^n - 4).
Cf. also permutations A267111, A267112.

(define (A265901 n) (A265901bi (A002260 n) (A004736 n)))
(define (A265901bi row col) (if (= 1 col) (A188163 row) (A087686 (+ 1 (A265901bi row (- col 1))))))

For the first column k=1, A(n,1) = A188163(n), for columns k > 1, A(n,k) = A087686(1+A(n,k-1)).

A162598 Ordinal transform of A265332.

Original entry on oeis.org

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

Offset: 1

Franklin T. Adams-Watters, Jul 07 2009

This is a fractal sequence.
It appears that each group of 2^k terms starts with 1 and ends with the remaining powers of two from 2^k down to 2^1.
From Antti Karttunen, Jan 09-12 2016: (Start)
This is ordinal transform of A265332, which is modified A051135 (with a(1) = 1, instead of 2). -  after Franklin T. Adams-Watters' original definition for this sequence.
A000079 (powers of 2) indeed gives the positions of ones in this sequence. This follows from the properties (3) and (4) of A004001 given on page 227 of Kubo & Vakil paper (page 3 of PDF), which together also imply the pattern observed above, more clearly represented as:
a(2) = 1.
a(3..4) = 2, 1.
a(6..8) = 4, 2, 1.
a(13..16) = 8, 4, 2, 1.
a(28..31) = 16, 8, 4, 2, 1.
etc.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..8192
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.
Index entries for Hofstadter-type sequences

Cf. A004001, A051135, A080677, A087686.
Row index of A265901, column index of A265903.
Cf. A265332 (corresponding other index).
Cf. A000079 (positions of ones).
Cf. A000225 (from the term 3 onward the positions of 2's).
Cf. A000325 (from its third term 5 onward the positions of 3's, which occur always as the last term before the next descending subsequence of powers of two).

terms = 100;
h[1] = 1; h[2] = 1;
h[n_] := h[n] = h[h[n - 1]] + h[n - h[n - 1]];
t = Array[h, 2*terms];
A051135 = Take[Transpose[Tally[t]][[2]], terms];
b[_] = 1;
a[n_] := a[n] = With[{t = If[n == 1, 1, A051135[[n]]]}, b[t]++];
Array[a, terms] (* Jean-François Alcover, Dec 19 2021, after Robert G. Wilson v in A051135 *)

Let b(1) = 1, b(n) = A051135(n) for n > 1. Then a(n) is the number of b(k) that equal b(n) for 1 <= k <= n: sum( 1, 1<=k<=n and a(k)=a(n) ).

If A265332(n) = 1, then a(n) = A004001(n), otherwise a(n) = a(A080677(n)-1) = a(n - A004001(n)). - Antti Karttunen, Jan 09 2016

Name amended by Antti Karttunen, Jan 09 2016

cp's OEIS Frontend

A267110 If A051135(n) = 1, then a(n) = A004001(n) - 1, otherwise a(n) = n - A004001(n).

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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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A087686 Elements of A004001 that repeat consecutively.

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A088359 Numbers which occur only once in A004001.

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A265903 Square array read by descending antidiagonals: A(1,k) = A188163(k), and for n > 1, A(n,k) = A087686(1+A(n-1,k)).

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A093879 First differences of A004001.

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A188163 Smallest m such that A004001(m) = n.