A226222 -id:A226222 - cp's OEIS frontend

A005185 Hofstadter Q-sequence: a(1) = a(2) = 1; a(n) = a(n-a(n-1)) + a(n-a(n-2)) for n > 2.

1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 8, 8, 8, 10, 9, 10, 11, 11, 12, 12, 12, 12, 16, 14, 14, 16, 16, 16, 16, 20, 17, 17, 20, 21, 19, 20, 22, 21, 22, 23, 23, 24, 24, 24, 24, 24, 32, 24, 25, 30, 28, 26, 30, 30, 28, 32, 30, 32, 32, 32, 32, 40, 33, 31, 38, 35, 33, 39, 40, 37, 38, 40, 39

Offset: 1

Simon Plouffe and N. J. A. Sloane, May 20 1991

Rate of growth is not known. In fact it is not even known if this sequence is defined for all positive n.
Roman Pearce, Aug 29 2014, has computed that a(n) exists for n <= 10^10. - N. J. A. Sloane
a(n) exists for n <= 3*10^10. - M. Eric Carr, Jul 02 2023

			a(18) = 11 because a(17) is 10 and a(16) is 9, so we take a(18 - 10) + a(18 - 9) = a(8) + a(9) = 5 + 6 = 11.

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 in Number Theory, Sect. E31.
D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 138.
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 and N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Altug Alkan, On a conjecture about generalized Q-recurrence, Open Mathematics, Vol. 16 (2018), pp. 1490-1500.
Altug Alkan, On a Generalization of Hofstadter's Q-Sequence: A Family of Chaotic Generational Structures, Complexity (2018) Article ID 8517125.
Altug Alkan, Nathan Fox, and Orhan Ozgur Aybar, On Hofstadter Heart Sequences, Complexity, 2017.
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.
Thomas Bloom, Problem 422, Erdős Problems.
Paul Bourke, Hofstadter "Q" Series.
Paul Bourke, Hofstadter "Q" Series. [Cached copy, pdf only, with permission]
Paul Bourke, Listen to first 300 terms of this sequence. [Cached copy from the Hofstadter "Q" web page, with permission]
Joseph Callaghan, John J. Chew III, and Stephen M. Tanny, On the behavior of a family of meta-Fibonacci sequences, SIAM Journal on Discrete Mathematics 18.4 (2005): 794-824. See p. 794.
Benoit Cloitre, Plot of a(n)/n - 1/2 for n=1 up to 2^19.
J.-P. Davalan, Douglas Hofstadter's sequences.
Jonathan H. B. Deane and Guido Gentile, A diluted version of the problem of the existence of the Hofstadter sequence, arXiv:2311.13854 [math.NT], 2023.
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.
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, Finding Linear-Recurrent Solutions to Hofstadter-Like Recurrences Using Symbolic Computation, arXiv:1609.06342 [math.NT], 2016.
Nathan Fox, A Slow Relative of Hofstadter's Q-Sequence, arXiv:1611.08244 [math.NT], 2016.
Nathan Fox, A New Approach to the Hofstadter Q-Recurrence, arXiv:1807.01365 [math.NT], 2018.
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, Some suspiciously simple sequences, Amer. Math. Monthly 93 (1986), 186-190; 94 (1987), 965; 96 (1989), 905.
Nick Hobson, Python program for this sequence.
D. R. Hofstadter, Eta-Lore. [Cached copy, with permission]
D. R. Hofstadter, Pi-Mu Sequences. [Cached copy, with permission]
D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991.
D. R. Hofstadter, Curious patterns and non-patterns in a family of meta-Fibonacci recursions, Lecture in Doron Zeilberger's Experimental Mathematics Seminar, Rutgers University, April 10 2014; Part 1, Part 2.
D. R. Hofstadter, Plot of first 100 million terms.
Abraham Isgur, Mustazee Rahman, and Stephen Tanny, Solving non-homogeneous nested recursions using trees, Annals of Combinatorics 17.4 (2013): 695-710. See p. 695. - _N. J. A. Sloane_, Apr 16 2014
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.
Piotr Miska, Bartosz Sobolewski, and Maciej Ulas, Binary sequences meet the Fibonacci sequence, arXiv:2412.11319 [math.NT], 2024. See p. 20.
K. Pinn, Order and chaos in Hofstadter's Q(n) sequence, Complexity, 4:3 (1999), 41-46.
K. Pinn, A chaotic cousin of Conway's recursive sequence, Experimental Mathematics, 9:1 (2000), 55-65.
K. Pinn, A Chaotic Cousin Of Conway's Recursive Sequence, arXiv:cond-mat/9808031 [cond-mat.stat-mech], 1998.
N. J. A. Sloane and Brady Haran, Amazing Graphs III, Numberphile video (2019).
I. Vardi, Email to N. J. A. Sloane, Jun. 1991.
A. M. M. Sharif Ullah, Surface Roughness Modeling Using Q-Sequence, Mathematical and Computational Applications, 2017.
Eric Weisstein's World of Mathematics, Hofstadter's Q-Sequence.
Wikipedia, Hofstadter sequence.
Pedro Zanetti, C++ code snippet for this sequence.
Index entries for sequences from "Goedel, Escher, Bach"
Index entries for Hofstadter-type sequences

Cf. A004001, A005206, A005374, A005375, A005378.
Cf. A005379, A070867, A226222, A239913, A284019.
Cf. A081827 (first differences).
Cf. A226244, A226245 (record values and where they occur).
See A244477 for a different start.

#include 
#define LIM 20
int Qa[LIM];
int Q(int n){if (n==1 || n==2){return 1;} else{return Qa[n-Qa[n-1]]+Qa[n-Qa[n-2]];}}
int main(){int i;printf("n\tQ\n");for(i=1; iGonzalo Ciruelos, Aug 01 2013

a005185 n = a005185_list !! (n-1)
a005185_list = 1 : 1 : zipWith (+)
   (map a005185 $ zipWith (-) [3..] a005185_list)
   (map a005185 $ zipWith (-) [3..] $ tail a005185_list)
-- Reinhard Zumkeller, Jun 02 2013, Sep 15 2011

I:=[1,1]; [n le 2 select I[n] else Self(n-Self(n-1))+Self(n-Self(n-2)): n in [1..90]]; // Vincenzo Librandi, Aug 08 2014

A005185 := proc(n) option remember;
    if n<=2 then 1
    elif n > procname(n-1) and n > procname(n-2) then
        RETURN(procname(n-procname(n-1))+procname(n-procname(n-2)));
    else
        ERROR(" died at n= ", n);
    fi; end proc;
# More generally, the following defines the Hofstadter-Huber sequence Q(r,s) - N. J. A. Sloane, Apr 15 2014
r:=1; s:=2;
a:=proc(n) option remember; global r,s;
if n <= s then  1
else
    if (a(n-r) <= n) and (a(n-s) <= n) then
    a(n-a(n-r))+a(n-a(n-s));
    else lprint("died with n =",n); return (-1);
    fi; fi; end;
[seq(a(n), n=1..100)];

a[1]=a[2]=1; a[n_]:= a[n]= a[n -a[n-1]] + a[n -a[n-2]]; Table[ a[n], {n,70}]

q:=proc(n) option remember; begin if n<=2 then 1 else q(n-q(n-1))+q(n-q(n-2)) end_if; end_proc: q(i)$i=1..100; // Zerinvary Lajos, Apr 03 2007

{a(n)= local(A); if(n<1, 0, A=vector(n,k,1); for(k=3, n, A[k]= A[k-A[k-1]]+ A[k-A[k-2]]); A[n])} /* Michael Somos, Jul 16 2007 */

from functools import lru_cache
@lru_cache(maxsize=None)
def a(n):
    if n < 3: return 1
    return a(n - a(n-1)) + a(n - a(n-2))
print([a(n) for n in range(1, 75)]) # Michael S. Branicky, Jul 26 2021

@CachedFunction
def a(n):
    if (n<3): return 1
    else: return a(n -a(n-1)) + a(n -a(n-2))
[a(n) for n in (1..70)] # G. C. Greubel, Feb 13 2020

(define q (lambda (n) (cond ( (eqv? n 0) 1) ( (eqv? n 1) 1) ( #t (+ (q (- n (q (- n 1)))) (q (- n (q (- n 2)))))))))

;; An implementation of memoization-macro definec can be found for example in: http://oeis.org/wiki/Memoization
(definec (A005185 n) (if (<= n 2) 1 (+ (A005185 (- n (A005185 (- n 1)))) (A005185 (- n (A005185 (- n 2)))))))
;; Antti Karttunen, Mar 22 2017

A046699 a(1) = a(2) = 1, a(n) = a(n - a(n-1)) + a(n-1 - a(n-2)) if n > 2.

Original entry on oeis.org

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

Offset: 1

R. K. Guy

Ignoring first term, this is the meta-Fibonacci sequence for s=0. - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)

Except for the first term, n occurs A001511(n) times. - Franklin T. Adams-Watters, Oct 22 2006

Sequence was proposed by Reg Allenby.
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. See Eq. (2).
Michael Doob, The Canadian Mathematical Olympiad & L'Olympiade Mathématique du Canada 1969-1993, Canadian Mathematical Society & Société Mathématique du Canada, Problem 5, 1990, pp. 212-213, 1993.
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.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
Altug Alkan, On a Generalization of Hofstadter's Q-Sequence: A Family of Chaotic Generational Structures, Complexity (2018) Article ID 8517125.
Joerg Arndt, Matters Computational (The Fxtbook)
Stephen M. Buckley, Wiring switches to more light bulbs, arXiv:2410.02460 [math.CO], 2024. See pp. 3, 12, 22.
Joseph Callaghan, John J. Chew III, and Stephen M. Tanny, On the behavior of a family of meta-Fibonacci sequences, SIAM Journal on Discrete Mathematics 18.4 (2005): 794-824. See p. 794. - _N. J. A. Sloane_, Apr 16 2014
M. Celaya and F. Ruskey, Morphic Words and Nested Recurrence Relations, arXiv preprint arXiv:1307.0153 [math.CO], 2013.
C. Deugau and F. Ruskey, Complete k-ary Trees and Generalized Meta-Fibonacci Sequences, J. Integer Seq., Vol. 12. [This is a later version than that in the GenMetaFib.html link]
C. Deugau and F. Ruskey, Complete k-ary Trees and Generalized Meta-Fibonacci Sequences
A. Erickson, A. Isgur, B. W. Jackson, F. Ruskey and S. M. Tanny, Nested recurrence relations with Conolly-like solutions, See Table 2.
Nathan Fox, A Slow Relative of Hofstadter's Q-Sequence, arXiv:1611.08244 [math.NT], 2016.
Nathan Fox, Trees, Fibonacci Numbers, and Nested Recurrences, Rutgers University Experimental Math Seminar, Mar 07, 2019.
Nathan Fox, Connecting Slow Solutions to Nested Recurrences with Linear Recurrent Sequences, arXiv:2203.09340 [math.CO], 2022.
The IMO Compendium, Problem 5, 22nd Canadian Mathematical Olympiad 1990.
Abraham Isgur, Mustazee Rahman, and Stephen Tanny, Solving non-homogeneous nested recursions using trees, Annals of Combinatorics 17.4 (2013): 695-710. See p. 695. - _N. J. A. Sloane_, Apr 16 2014
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.
B. Jackson and F. Ruskey, Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes
B. Jackson and F. Ruskey, Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes, Electronic Journal of Combinatorics, 13 (2006), #R26, 13 pages.
Oliver Kullmann and Xishun Zhao, Parameters for minimal unsatisfiability: Smarandache primitive numbers and full clauses, arXiv preprint, arXiv:1505.02318 [cs.DM], 2015.
Thomas M. Lewis and Fabian Salinas, Optimal pebbling of complete binary trees and a meta-Fibonacci sequence, arXiv:2109.07328 [math.CO], 2021.
Ramin Naimi and Eric Sundberg, A Combinatorial Problem Solved by a Meta-Fibonacci Recurrence Relation, arXiv:1902.02929 [math.CO], 2019.
Index entries for Hofstadter-type sequences.
Index to sequences related to Olympiads.

Cf. A001511, A005185, A005187, A007843, A055938, A079559, A080578, A101925, A182105, A213714, A226222, A234016, A275363, A324473, A324475, A324477.

Callaghan et al. (2005)'s sequences T_{0,k}(n) for k=1 through 7 are A000012, A046699, A046702, A240835, A241154, A241155, A240830.

a046699 n = a046699_list !! (n-1)
a046699_list = 1 : 1 : zipWith (+) zs (tail zs) where
   zs = map a046699 $ zipWith (-) [2..] a046699_list
-- Reinhard Zumkeller, Jan 02 2012

[ n le 2 select 1 else Self(n - Self(n-1)) + Self(n-1 -Self(n-2)):n in [1..80]]; // Marius A. Burtea, Oct 17 2019

a := proc(n) option remember; if n <= 1 then return 1 end if; if n <= 2 then return 2 end if; return add(a(n - i + 1 - a(n - i)), i = 1 .. 2) end proc # Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)
a := proc(n) option remember; if n <= 2 then 1 else a(n - a(n-1)) + a(n-1 - a(n-2)); fi; end; # N. J. A. Sloane, Apr 16 2014

a[n_] := (k = 1; While[ !Divisible[(2*++k)!, 2^(n-1)]]; k); a[1] = a[2] = 1; Table[a[n], {n, 1, 72}] (* Jean-François Alcover, Oct 06 2011, after Benoit Cloitre *)
CoefficientList[ Series[1 + x/(1 - x)*Product[1 + x^(2^n - 1), {n, 6}], {x, 0, 80}], x] (* or *)
a[1] = a[2] = 1; a[n_] := a[n] = a[n - a[n - 1]] + a[n - 1 - a[n - 2]]; Array[a, 80] (* Robert G. Wilson v, Sep 08 2014 *)

a[1]:1$
a[2]:1$
a[n]:=a[n-a[n-1]]+a[n-1-a[n-2]]$
makelist(a[n],n,2,60); /* Martin Ettl, Oct 29 2012 */

a(n)=if(n<0,1,s=1;while((2*s)!%2^(n-1)>0,s++);s) \\ Benoit Cloitre, Jan 19 2007

from sympy import factorial
def a(n):
    if n<3: return 1
    s=1
    while factorial(2*s)%(2**(n - 1))>0: s+=1
    return s
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 11 2017, after Benoit Cloitre

First differences seem to be A079559. - Vladeta Jovovic, Nov 30 2003. This is correct and not too hard to prove, giving the generating function x + x^2(1+x)(1+x^3)(1+x^7)(1+x^15).../(1-x). - Paul Boddington, Jul 30 2004
G.f.: x + x^2/(1-x) * Product_{n=1}^{infinity} (1 + x^(2^n-1)). - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)
For n>=1, a(n)=w(n-1) where w(n) is the least k such that 2^n divides (2k)!. - Benoit Cloitre, Jan 19 2007
Conjecture: a(n+1) = a(n) + A215530(a(n) + n) for all n > 0. - Velin Yanev, Oct 17 2019
From Bernard Schott, Dec 03 2021: (Start)
a(n) <= a(n+1) <=  a(n) +1.
For n > 1, if a(n) is odd, then a(n+1) = a(n) + 1.
a(2^n+1) = 2^(n-1) + 1 for n > 0.
Results coming from the 5th problem proposed during the 22nd Canadian Mathematical Olympiad in 1990 (link IMO Compendium and Doob reference). (End)

A070867 a(1) = a(2) = 1; a(n) = a(n-1 - a(n-1)) + a(n-1 - a(n-2)).

Original entry on oeis.org

1, 1, 2, 2, 2, 4, 3, 4, 4, 4, 8, 5, 5, 8, 8, 6, 8, 12, 8, 11, 9, 9, 10, 13, 16, 9, 12, 20, 10, 12, 23, 12, 15, 21, 13, 17, 18, 19, 19, 22, 21, 19, 19, 26, 28, 16, 24, 33, 21, 26, 23, 36, 16, 26, 39, 16, 30, 33, 36, 19, 34, 31, 43, 23, 30, 32, 38, 23, 40, 26, 38, 43, 31, 31, 38, 44

Offset: 1

N. J. A. Sloane, May 19 2002

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 129.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Nick Hobson, Python program for this sequence
Eric Weisstein's World of Mathematics, Wolfram Sequences
Index entries for Hofstadter-type sequences

Cf. A005185, A226222.

a070867 n = a070867_list !! (n-1)
a070867_list = 1 : 1 : zipWith (+)
   (map a070867 $ zipWith (-) [2..] a070867_list)
   (map a070867 $ zipWith (-) [2..] $ tail a070867_list)
-- Reinhard Zumkeller, May 31 2013

a[1]=a[2]=1; a[n_]:= a[n]= a[n-1 -a[n-1]] + a[n-1 -a[n-2]]; Table[a[n], {n, 80}]

@CachedFunction
def a(n): # A070867
    if (n<3): return 1
    else: return  a(n-1 -a(n-1)) + a(n-1 -a(n-2))
[a(n) for n in (1..80)] # G. C. Greubel, Mar 28 2022

a(n) = a(n-1 - a(n-1)) + a(n-1 - a(n-2)), with a(1) = a(2) = 1.

More terms from John W. Layman, May 21 2002

Erroneous comma in sequence corrected by Nick Hobson, Feb 17 2007

A063892 a(1) = a(2) = a(3) = 1, a(n) = a(n-a(n-2))+a(n-a(n-3)) for n>3.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 5, 3, 3, 9, 6, 4, 7, 12, 9, 5, 7, 10, 16, 16, 10, 12, 12, 16, 14, 21, 13, 17, 8, 14, 24, 26, 19, 12, 8, 23, 22, 23, 12, 17, 18, 28, 35, 26, 16, 22, 34, 47, 22, 6, 10, 36, 69, 36

Offset: 1

Henry Bottomley, Aug 29 2001

Undefined for n>54 since a(53)=69>=55.

			a(7) = a(7-a(5))+a(7-a(4)) = a(7-4)+a(7-2) = a(3)+a(5) = 1+4 = 5.

Cf. A005185, A046700, A063882, A226222.

See A064714 for a variant which does not die.

#Q(r,s) with initial values 1,1,1,..., from N. J. A. Sloane, Apr 15 2014
r:=2; s:=3;
a:=proc(n) option remember; global r,s;
if n <= s then 1
else
    if (a(n-r) <= n) and (a(n-s) <= n) then
    a(n-a(n-r))+a(n-a(n-s));
    else lprint("died with n =",n); return (-1);
    fi;
fi; end;
[seq(a(n),n=1..54)];

A284511 a(1) = 1, a(2) = 2, a(3) = 2; a(n) = a(a(n-1)-1) + a(n-a(n-2)) for n > 3.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 5, 7, 8, 7, 7, 8, 8, 8, 9, 12, 14, 12, 11, 15, 15, 13, 14, 15, 15, 15, 16, 16, 16, 16, 17, 21, 23, 23, 23, 23, 23, 24, 25, 26, 27, 27, 27, 29, 28, 29, 27, 26, 28, 30, 30, 29, 30, 31, 31, 31, 31, 32, 32, 32, 32, 32, 33, 38, 44, 43, 41, 40, 40, 40, 41, 46, 50

Offset: 1

Altug Alkan, Mar 28 2017

nonn

			a(4) = 3 because a(4) = a(a(3) - 1) + a(4 - a(2)) = a(1) + a(2) = 3.

Altug Alkan, Table of n, a(n) for n = 1..20000
Altug Alkan, Scatterplot of A284511
Altug Alkan, Scatterplot of 2*a(n)-n

Cf. A004001, A005185, A005229, A055748, A070867, A226222.

a:= proc(n) option remember;
  procname(procname(n-1)-1)+procname(n-procname(n-2))
end proc:
a(1):= 1: a(2):= 2: a(3):= 2:
map(a, [$1..100]); # Robert Israel, Mar 28 2017

a[1]=1; a[2]=a[3]=2; a[n_] := a[n] = a[a[n-1]-1] + a[n-a[n-2]]; Array[a, 74] (* Giovanni Resta, Mar 28 2017 *)

a=vector(1000); a[1]=1; a[2]=a[3]=2; for(n=4, #a, a[n]=a[a[n-1]-1]+a[n-a[n-2]]); a

@CachedFunction
def a(n): # A284511
    if (n<4): return 2 - bool(n==1)
    else: return  a(a(n-1)-1) + a(n-a(n-2))
[a(n) for n in (1..80)] # G. C. Greubel, Mar 28 2022

a(n) = a(a(n-1)-1) + a(n-a(n-2)), with a(1) = 1, a(2) = 2, a(3) = 2.

cp's OEIS Frontend

A005185 Hofstadter Q-sequence: a(1) = a(2) = 1; a(n) = a(n-a(n-1)) + a(n-a(n-2)) for n > 2.

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A046699 a(1) = a(2) = 1, a(n) = a(n - a(n-1)) + a(n-1 - a(n-2)) if n > 2.

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A070867 a(1) = a(2) = 1; a(n) = a(n-1 - a(n-1)) + a(n-1 - a(n-2)).

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A063892 a(1) = a(2) = a(3) = 1, a(n) = a(n-a(n-2))+a(n-a(n-3)) for n>3.

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A284511 a(1) = 1, a(2) = 2, a(3) = 2; a(n) = a(a(n-1)-1) + a(n-a(n-2)) for n > 3.

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A284520 a(1) = 1, a(2) = a(3) = a(4) = 2; a(n) = a(a(n-1)-1) + a(n-a(n-3)-1) for n > 4.

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