A088359 -id:A088359 - cp's OEIS frontend

A051135 a(n) = number of times n appears in the Hofstadter-Conway $10000 sequence A004001.

2, 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

Offset: 1

Robert Lozyniak (11(AT)onna.com)

If the initial 2 is changed to a 1, the resulting sequence (A265332) has the property that if all 1's are deleted, the remaining terms are the sequence incremented. - Franklin T. Adams-Watters, Oct 05 2006
a(A088359(n)) = 1 and a(A087686(n)) > 1; first differences of A188163. - Reinhard Zumkeller, Jun 03 2011
From Robert G. Wilson v, Jun 07 2011: (Start)
a(k)=1 for k = 3, 5, 6, 9, 10, 11, 13, 17, 18, 19, 20, 22, 23, 25, 28, ..., ; (A088359)
a(k)=2 for k = 1, 2, 7, 12, 14, 21, 24, 26, 29, 38, 42, 45, 47, 51, 53, ..., ; (1 followed by A266109)
a(k)=3 for k = 4, 15, 27, 30, 48, 54, 57, 61, 86, 96, 102, 105, 112, ..., ; (A267103)
a(k)=4 for k = 8, 31, 58, 62, 106, 116, 120, 125, 192, 212, 222, 226, ..., ;
a(k)=5 for k = 16, 63, 121, 126, 227, 242, 247, 253, 419, 454, 469, ..., ;
a(k)=6 for k = 32, 127, 248, 254, 475, 496, 502, 509, 894, 950, 971, ..., ;
a(k)=7 for k = 64, 255, 503, 510, 978, 1006, 1013, 1021, 1872, 1956, ..., ;
a(k)=8 for k = 128, 511, 1014, 1022, 1992, 2028, 2036, 2045, 3864, ..., ;
a(k)=9 for k = 256, 1023, 2037, 2046, 4029, 4074, 4083, 4093, 7893, ..., ;
a(k)=10 for k = 512, 2047, 4084, 4094, 8113, 8168, 8178, 8189, ..., . (End)
Compare above to array A265903. - Antti Karttunen, Jan 18 2016

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.
Eric Weisstein's World of Mathematics, Hofstadter-Conway $10,000 Sequence.
Wikipedia, Hofstadter sequence
Index entries for Hofstadter-type sequences

Cf. A004001, A087686, A093879, A188163, A266109, A267103 and array A265903.
Cf. A088359 (positions of ones).
Cf. A265332 (essentially the same sequence, but with a(1) = 1 instead of 2).

import Data.List (group)
a051135 n = a051135_list !! (n-1)
a051135_list = map length $ group a004001_list
-- Reinhard Zumkeller, Jun 03 2011

nmax:=200;
h:=[n le 2 select 1 else Self(Self(n-1)) + Self(n - Self(n-1)): n in [1..5*nmax]]; // h = A004001
A188163:= function(n)
   for j in [1..3*nmax+1] do
       if h[j] eq n then return j; end if;
   end for;
end function;
A051135:= func< n | A188163(n+1) - A188163(n) >;
[A051135(n): n in [1..nmax]]; // G. C. Greubel, May 20 2024

a[1]:=1: a[2]:=1: for n from 3 to 300 do a[n]:=a[a[n-1]]+a[n-a[n-1]] od: A:=[seq(a[n],n=1..300)]:for j from 1 to A[nops(A)-1] do c[j]:=0: for n from 1 to 300 do if A[n]=j then c[j]:=c[j]+1 else fi od: od: seq(c[j],j=1..A[nops(A)-1]); # Emeric Deutsch, Jun 06 2006

a[1] = 1; a[2] = 1; a[n_] := a[n] = a[a[n - 1]] + a[n - a[n - 1]]; t = Array[a, 250]; Take[ Transpose[ Tally[t]][[2]], 105] (* Robert G. Wilson v, Jun 07 2011 *)

@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+1):
        if h(j)==n: return j
def A051135(n): return A188163(n+1) - A188163(n)
[A051135(n) for n in range(1,201)] # G. C. Greubel, May 20 2024

(define (A051135 n) (- (A188163 (+ 1 n)) (A188163 n))) ;; Antti Karttunen, Jan 18 2016

From Antti Karttunen, Jan 18 2016: (Start)
a(n) = A188163(n+1) - A188163(n). [after Reinhard Zumkeller's Jun 03 2011 comment above]
Other identities:
a(n) = 1 if and only if A093879(n-1) = 1. [See A188163 for a reason.]
(End)

More terms from Jud McCranie

Added links (in parentheses) to recently submitted related sequences - Antti Karttunen, Jan 18 2016

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).

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)).

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

Original entry on oeis.org

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.]

A267108 a(n) = A000120(A267111(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 16 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000120, A004001, A070939, A088359, A265332, A267109, A267110, A267111.

a(1) = 1; for n > 1, if A265332(n) = 1 [when n is one of the terms of A088359], a(n) =  1 + a(A004001(n)-1), otherwise a(n) = a(n-A004001(n)).
a(n) = A000120(A267111(n)).
Other identities. For all n >= 1:
a(n) = A070939(n) - A267109(n).

A267109 a(n) = A080791(A267111(n)).

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 1, 3, 2, 1, 0, 2, 1, 1, 2, 4, 3, 2, 1, 0, 3, 2, 1, 2, 1, 1, 3, 2, 2, 2, 3, 5, 4, 3, 2, 1, 0, 4, 3, 2, 1, 3, 2, 1, 2, 1, 1, 4, 3, 2, 3, 2, 2, 3, 2, 2, 2, 4, 3, 3, 3, 3, 4, 6, 5, 4, 3, 2, 1, 0, 5, 4, 3, 2, 1, 4, 3, 2, 1, 3, 2, 1, 2, 1, 1, 5, 4, 3, 2, 4, 3, 2, 3, 2, 2, 4, 3, 2, 3, 2, 2, 3, 2, 2, 2, 5, 4, 3, 4, 3, 3, 4, 3, 3, 3, 4, 3, 3, 3, 3

Offset: 1

Antti Karttunen, Jan 16 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A004001, A070939, A080791, A088359, A265332, A267108, A267110, A267111.

a(1) = 0; for n > 1, if A265332(n) = 1 [when n is one of the terms of A088359], a(n) =  a(A004001(n)-1), otherwise 1 + a(n) = a(n-A004001(n)).
a(n) = A080791(A267111(n)).
Other identities. For all n >= 1:
a(n) = A070939(n) - A267108(n).

A118179 Numbers which occur only once in A005229.

Original entry on oeis.org

2, 4, 5, 8, 9, 11, 13, 14, 15, 18, 21, 22, 23, 25, 27, 28, 31, 32, 33, 34, 35, 38, 40, 41, 42, 44, 47, 48, 49, 50, 51, 52, 55, 56, 57, 60, 61, 62, 65, 67, 68, 70, 71, 72, 73, 74, 75, 76, 79, 82, 83, 84, 86, 88, 89, 90, 91, 96, 98, 99, 100, 101, 103, 104, 105, 106, 107, 108

Offset: 1

Klaus Brockhaus, Apr 13 2006

nonn

			1, 3, 6 and 7 occur twice in A005229, so these numbers are not in the sequence.

Cf. A005229, A087758, A088359.

terms = 70; Clear[b, f]; b[1] = b[2] = 1; b[n_] := b[n] = b[b[n - 2]] + b[n - b[n - 2]]; f[max_] := f[max] = PadRight[Select[Tally[Array[b, max]], #[[2]] == 1 &][[All, 1]], terms]; f[max = terms]; f[max = max + terms]; While[f[max] != f[max - terms], max = max + terms]; A118179 = f[max](* Jean-François Alcover, Oct 12 2017 *)

{m=156;v=vector(m);v[1]=1;v[2]=1;for(n=3,m,v[n]=v[v[n-2]]+v[n-v[n-2]]); i=1;k=1;while(i<=m,c=0;while(i<=m&&v[i]==k,c++;i++);if(i<=m&&c==1,print1(k,","));k++)}

cp's OEIS Frontend

A051135 a(n) = number of times n appears in the Hofstadter-Conway $10000 sequence A004001.

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

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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)).

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

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A267108 a(n) = A000120(A267111(n)).

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A267109 a(n) = A080791(A267111(n)).

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A118179 Numbers which occur only once in A005229.

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