A073869 -id:A073869 - cp's OEIS frontend

A090909 Terms a(k) of A073869 for which a(k-1) = a(k), and a(k) and a(k+1) are distinct.

2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, 36, 39, 41, 44, 47, 49, 52, 54, 57, 60, 62, 65, 68, 70, 73, 75, 78, 81, 83, 86, 89, 91, 94, 96, 99, 102, 104, 107, 109, 112, 115, 117, 120, 123, 125, 128, 130, 133, 136, 138, 141, 143, 146, 149, 151, 154, 157, 159, 162

Offset: 1

Amarnath Murthy, Dec 14 2003

nonn

Is this the same as A001950? - Alec Mihailovs (alec(AT)mihailovs.com), Jul 23 2007
Identical to n + A066096(n)? - Ed Russell (times145(AT)hotmail.com), May 09 2009
From Michel Dekking, Dec 18 2024: (Start)
Proof of Mihailovs's conjecture: This follows immediately from the result in my 2023 paper in JIS that A073869 is equal to Hofstadter’s G-sequence A005206, and my recent comment in A005206 on the pairs of duplicate values in A005206.
The answer to Russell’s question is well-known, and also Detlef’s formula is well-known.
Originally, this sequence was given the name “Terms a(k) of A073869 for which a(k-1), a(k) and a(k+1) are distinct.’’ These are the triples (1,2,3),(4,5,6),(6,7,8), (9,10,11), ... occurring at  k = 3, k = 8, k = 11, k = 16,... in A005206. Note that if a duplicate pair (a(m-1), a(m)) is followed directly by another duplicate pair, then a(m-3), a(m-2) and a(m-1) are distinct, and only so. This corresponds to the block 00 occurring in the Fibonacci word obtained by projecting A005206 on the Fibonacci word (see Corollary in my recent comment in A005206). These occurrences are at the Wythoff AB numbers A003623 according to Wolfdieter Lang’s comment in A003623. Conclusion: the sequence of terms a(k) of A073869 for which a(k-1), a(k), and a(k+1) are distinct is given by the Wythoff AB-numbers. (End)

			A073869 = A005206 = 0,1,1,2,3,3,4,4,5,6,6,... The pair (1,1) occurs at k = 2.

G. C. Greubel, Table of n, a(n) for n = 0..10000
F. M. Dekking, On Hofstadter's G-sequence, Journal of Integer Sequences 26 (2023), Article 23.9.2, 1-11.

Cf. A001622, A001950, A002251, A005206, A073869, A090908.
Cf. A004919, A004920, A004921, A004922, A004923, A004924, A004925.
Cf. A004926, A004927, A004928, A004929, A004930, A004931, A004932.
Cf. A004933, A004934, A004935, A004976, A066096.

[Floor((3+Sqrt(5))*n/2): n in [0..80]]; // G. C. Greubel, Sep 12 2023

(* First program *)
A002251= Fold[Append[#1, #2 Ceiling[#2/GoldenRatio] -Total[#1]] &, {1}, Range[2, 500]] - 1; (* Birkas Gyorgy's code of A019444, modified *)
A090909= Join[{0}, Select[Partition[A002251, 2, 1], #[[2]] > #[[1]] &][[All, 2]]] (* G. C. Greubel, Sep 12 2023 *)
(* Second program *)
Floor[GoldenRatio^2*Range[0,80]] (* G. C. Greubel, Sep 12 2023 *)

[floor(golden_ratio^2*n) for n in range(81)] # G. C. Greubel, Sep 12 2023

a(n) = floor(phi^2*n), where phi = (1+sqrt(5))/2. - Gary Detlefs, Mar 10 2011

More terms from R. J. Mathar, Sep 29 2017

Name corrected by Michel Dekking, Dec 13 2024

A090908 Terms a(k) of A073869 for which a(k) is distinct from a(k-1).

Original entry on oeis.org

1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 25, 27, 29, 30, 32, 33, 35, 37, 38, 40, 42, 43, 45, 46, 48, 50, 51, 53, 55, 56, 58, 59, 61, 63, 64, 66, 67, 69, 71, 72, 74, 76, 77, 79, 80, 82, 84, 85, 87, 88, 90, 92, 93, 95, 97, 98, 100, 101, 103, 105, 106, 108, 110, 111

Offset: 1

Amarnath Murthy, Dec 14 2003

nonn

Is this the same as A000201?

The answer is yes: see A090909. - Michel Dekking, Dec 19 2024

			A073869 = A005206 = 0,1,1,2,3,3,4,4,5,6,6,... The pair (1,2) occurs at k = 3.

F. M. Dekking, On Hofstadter's G-sequence, Journal of Integer Sequences 26 (2023), Article 23.9.2, 1-11.

Cf. A002251, A005206, A073869, A090909, A000201.

See A000201.

More terms from R. J. Mathar, Sep 29 2017

Name corrected by Michel Dekking, Dec 19 2024

A090638 Duplicate of A073869.

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 17

Offset: 1

dead

A005206 Hofstadter G-sequence: a(0) = 0; a(n) = n - a(a(n-1)) for n > 0.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Rule for finding n-th term: a(n) = An, where An denotes the Fibonacci antecedent to (or right shift of) n, which is found by replacing each F(i) in the Zeckendorf expansion (obtained by repeatedly subtracting the largest Fibonacci number you can until nothing remains) by F(i-1) (A1=1). For example: 58 = 55 + 3, so a(58) = 34 + 2 = 36. - Diego Torres (torresvillarroel(AT)hotmail.com), Nov 24 2002
From Albert Neumueller (albert.neu(AT)gmail.com), Sep 28 2006: (Start)
A recursively built tree structure can be obtained from the sequence (see Hofstadter, p. 137):
  14  15  16  17  18  19  20  21
    \ /   /     \ /     \ /   /
     9  10      11      12  13
      \ /       /         \ /
       6       7           8
        \     /           /
         \   /           /
          \ /           /
           4           5
            \         /
             \       /
              \     /
               \   /
                \ /
                 3
                /
               2
            \ /
             1
To construct the tree: node n is connected with the node a(n) below
     n
    /
  a(n)
For example, since a(7) = 4:
    7
   /
  4
If the nodes of the tree are read from bottom to top, left to right, one obtains the positive integers: 1, 2, 3, 4, 5, 6, ... The tree has a recursive structure, since the construct
      /
     x
  \ /
   x
can be repeatedly added on top of its own ends, to construct the tree from its root: e.g.,
              /
             x
      /   \ /
     x     x
  \ /     /
   x     x
    \   /
     \ /
      x
When moving from a node to a lower connected node, one is moving to the parent. Parent node of n: floor((n+1)/tau). Left child of n: floor(tau*n). Right child of n: floor(tau*(n+1))-1 where tau=(1+sqrt(5))/2. (See the Sillke link.)
(End)
The number n appears A001468(n) times; A001468(n) = floor((n+1)*Phi) - floor(n*Phi) with Phi = (1 + sqrt 5)/2. - Philippe Deléham, Sep 22 2005
Number of positive Wythoff A-numbers A000201 not exceeding n. - N. J. A. Sloane, Oct 09 2006
Number of positive Wythoff B-numbers < A000201(n+1). - N. J. A. Sloane, Oct 09 2006
From Bernard Schott, Apr 23 2022: (Start)
Properties coming from the 1st problem proposed during the 45th Czech and Slovak Mathematical Olympiad in 1996 (see IMO Compendium link):
-> a(n) >= a(n-1) for any positive integer n,
-> a(n) - a(n-1) belongs to {0,1},
-> No integer n exists such that a(n-1) = a(n) = a(n+1). (End)
For n >= 1, find n in the Wythoff array (A035513). a(n) is the number that precedes n in its row, using the preceding column of the extended Wythoff array (A287870) if n is at the start of the (unextended) row. - Peter Munn, Sep 17 2022
See my 2023 publication on Hofstadter's G-sequence for a proof of the equality of (a(n)) with the sequence A073869. - Michel Dekking, Apr 28 2024
From Michel Dekking, Dec 16 2024: (Start)
Focus on the pairs of duplicate values, i.e., the pairs (a(n-1),a(n)) with a(n-1) = a(n). Directly from Theorem 1 in Kimberling and Stolarsky (2016) one derives that the m-th pair of duplicate values (a(n-1),a(n)) occurs at n = U(m), where U = 2,5,7,10,... is the upper Wythoff sequence. For example, (3,3) is the second pair, and occurs at U(2) = 5.
This property can be used to give a simple construction for (a(n)) -- ignoring the superfluous a(0) = 0.
Let 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,... be the sequence of positive natural numbers. Double all the elements of the lower Wythoff sequence (L(n)) = 1,3,4,6,8,9,11,....:
     (x(n)) := 1,1, 2, 3,3, 4,4, 5, 6,6, 7, 8,8, 9,9, 10,....
Claim: the result is (a(n)). This follows since because of the doubling, the m-th pair of duplicate values (a(n-1),a(n)) occurs in x at n = L(m) + m = U(m). The second equality by a well-known formula.
It follows from this by Theorem 1 of K&S, that a(n-1) = x(n-1), and a(n) = x(n) if n = U(m), for all m. But since L and U are complementary sequences, a(n) = x(n) will also hold if n = L(k), for all k. For example, L(4) = 6, and a(6) = x(6) = 4.
Corollary: for n >= 2 replace every pair of duplicate values (a(n-1),a(n)) by 0, and all the remaining elements of (a(n)) by 1. Then the result is the Fibonacci word 0,1,0,0,1,0,1,0,0... This is implied by the fact that L gives the positions of the 0s, and U the position of the 1's in the Fibonacci word. (End)
For all n >= 0, a(n) <= A005374(n), as proved in Letouzey-Li-Steiner link. Last equality occurs at n = 12, while a(n) < A005374(n) afterwards. - Pierre Letouzey, Feb 20 2025

D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 137.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane and T. D. Noe, Table of n, a(n) for n = 0..20000 (the first 1000 terms were found by T. D. Noe)
L. Carlitz, Fibonacci Representations, Fibonacci Quarterly, volume 6, number 4, October 1968, pages 193-220. a(n) = e(n) at equation 1.10 or 2.11 and see equation 3.8 recurrence.
M. Celaya and F. Ruskey, Morphic Words and Nested Recurrence Relations, arXiv preprint arXiv:1307.0153 [math.CO], 2013.
M. Celaya and F. Ruskey, Another Property of Only the Golden Ratio, American Mathematical Monthly, Problem 11651, solutions volume 121, number 6, June-July 2014, pages 549-556.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
F. M. Dekking, On Hofstadter's G-sequence, Journal of Integer Sequences 26 (2023), Article 23.9.2, 1-11.
Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, No. 1 (February 2012), pp. 11-18.
D. Gault and M. Clint, "Curiouser and curiouser" said Alice. Further reflections on an interesting recursive function, Internat. J. Computer Math., 26 (1988), 35-43. Also annotated scanned copy.
Martin Griffiths, A formula for an infinite family of Fibonacci-word sequences, Fib. Q., 56 (2018), 75-80.
H. W. Gould, J. B. Kim and V. E. Hoggatt, Jr., Sequences associated with t-ary coding of Fibonacci's rabbits, Fib. Quart., 15 (1977), 311-318.
Vincent Granville and Jean-Paul Rasson, A strange recursive relation, J. Number Theory 30 (1988), no. 2, 238--241. MR0961919(89j:11014).
J. Grytczuk, Another variation on Conway's recursive sequence, Discr. Math. 282 (2004), 149-161.
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
The IMO Compendium, Problem 1, 45th Czech and Slovak Mathematical Olympiad 1996.
Clark Kimberling and K. B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, 123 (No. 2, 2016), 267-273.
Pierre Letouzey, Hofstadter's problem for curious readers, Technical Report, 2015.
Pierre Letouzey, S. Li and W. Steiner, Pointwise order of generalized Hofstadter functions G,H and beyond, arXiv:2410.00529 [cs.DM], 2024.
Pierre Letouzey, Generalized Hofstadter functions G,H and beyond: numeration systems and discrepancy arXiv:2502.12615 [cs.DM], 2025.
Mustazee Rahman, A Combinatorial interpretation of Hofstadter's G-sequence, arXiv:1105.1718 [math.CO], 2011-2013.
B. Schoenmakers, A tight lower bound for top-down skew heaps, Information Processing Letters, 61(5): 279-284, 14 March 1997.
Torsten Sillke, Floor Recurrences
Th. Stoll, On Hofstadter's married functions, Fib. Q., 46/47 (2008/2009), 62-67. - _N. J. A. Sloane_, May 30 2009
Eric Weisstein's World of Mathematics, Hofstadter G-Sequence
Wikipedia, Hofstadter sequence
Index entries for Hofstadter-type sequences
Index entries for sequences from "Goedel, Escher, Bach"
Index to sequences related to Olympiads.

Apart from initial terms, same as A060143. Cf. A123070.
a(n):=Sum{k=1..n} h(k), n >= 1, with h(k):= A005614(k-1) and a(0):=0.
Cf. A060144, A019446, A072649, A213676, A000201.
Cf. A035513, A287870.

a005206 n = a005206_list !! n
a005206_list = 0 : zipWith (-) [1..] (map a005206 a005206_list)
-- Reinhard Zumkeller, Feb 02 2012, Aug 07 2011

a005206 = sum . zipWith (*) a000045_list . a213676_row . a000201 . (+ 1)
-- Reinhard Zumkeller, Mar 10 2013

[Floor((n+1)*(1+Sqrt(5))/2)-n-1: n in [0..80]]; // Vincenzo Librandi, Nov 19 2016

H:=proc(n) option remember; if n=0 then 0 elif n=1 then 1 else n-H(H(n-1)); fi; end proc: seq(H(n),n=0..76);

a[0] = 0; a[n_] := a[n] = n - a[a[n - 1]]; Array[a, 77, 0]
(* Second program: *)
Fold[Append[#1, #2 - #1[[#1[[#2]] + 1 ]] ] &, {0}, Range@ 76] (* Michael De Vlieger, Nov 13 2017 *)

first(n)=my(v=vector(n)); v[1]=1; for(k=2,n, v[k]=k-v[v[k-1]]); concat(0,v) \\ Charles R Greathouse IV, Sep 02 2015

from math import isqrt
def A005206(n): return (n+1+isqrt(5*(n+1)**2)>>1)-n-1 # Chai Wah Wu, Aug 09 2022

a(n) = floor((n+1)*tau) - n - 1 = A000201(n+1)-n-1, where tau = (1+sqrt(5))/2; or a(n) = floor(sigma*(n+1)) where sigma = (sqrt(5)-1)/2.
a(0)=0, a(1)=1, a(n) = n - a(floor(n/tau)). - Benoit Cloitre, Nov 27 2002
a(n) = A019446(n) - 1. - Reinhard Zumkeller, Feb 02 2012
a(n) = n - A060144(n+1). - Reinhard Zumkeller, Apr 07 2012
a(n) = Sum_{k=1..A072649(m)} A000045(m)*A213676(m,k): m=A000201(n+1). - Reinhard Zumkeller, Mar 10 2013
From Pierre Letouzey, Sep 09 2015: (Start)
a(n + a(n)) = n.
a(n) + a(a(n+1) - 1) = n.
a(0) = 0, a(n+1) = a(n) + d(n) and d(0) = 1, d(n+1)=1-d(n)*d(a(n)). (End)
a(n) = A293688(n)/(n+1) for n >= 0 (conjectured). - Enrique Navarrete, Oct 15 2017
A generalization of Diego Torres's 2002 comment as a formula: if n = Sum_{i in S} A000045(i+1), where S is a set of positive integers, then a(n) = Sum_{i in S} A000045(i). - Peter Munn, Sep 28 2022
Conjectures from Chunqing Liu, Aug 01 2023: (Start)
a(A000201(n)-1) = n-1.
a(A001950(n)-1) = a(A001950(n)) = A000201(n). (End)

a(0) = 0 added in the Name by Bernard Schott, Apr 23 2022

A002251 Start with the nonnegative integers; then swap L(k) and U(k) for all k >= 1, where L = A000201, U = A001950 (lower and upper Wythoff sequences).

Original entry on oeis.org

0, 2, 1, 5, 7, 3, 10, 4, 13, 15, 6, 18, 20, 8, 23, 9, 26, 28, 11, 31, 12, 34, 36, 14, 39, 41, 16, 44, 17, 47, 49, 19, 52, 54, 21, 57, 22, 60, 62, 24, 65, 25, 68, 70, 27, 73, 75, 29, 78, 30, 81, 83, 32, 86, 33, 89, 91, 35, 94, 96, 37, 99, 38, 102, 104, 40, 107, 109

Offset: 0

Michael Kleber

(n,a(n)) are Wythoff pairs: (0,0), (1,2), (3,5), (4,7), ..., where each difference occurs once.
Self-inverse when considered as a permutation or function, i.e., a(a(n)) = n. - Howard A. Landman, Sep 25 2001
If the offset is 1, the sequence can also be obtained by rearranging the natural numbers so that sum of n terms is a multiple of n, or equivalently so that the arithmetic mean of the first n terms is an integer. - Amarnath Murthy, Aug 16 2002
For n = 1, 2, 3, ..., let p(n)=least natural number not already an a(k), q(n) = n + p(n); then a(p(n)) = q(n), a(q(n)) = p(n). - Clark Kimberling
Also, indices of powers of 2 in A086482. - Amarnath Murthy, Jul 26 2003
There is a 7-state Fibonacci automaton (see a002251_1.pdf) that accepts, in parallel, the Zeckendorf representations of n and a(n). - Jeffrey Shallit, Jul 14 2023

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 76.

T. D. Noe, Table of n, a(n) for n = 0..10000
Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 18.
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, 43 pages, no date, unpublished.
Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, unpublished, no date [Cached copy, with permission]
Alex Meadows and B. Putman, A New Twist on Wythoff's Game, arXiv preprint arXiv:1606.06819 [math.CO], 2016.
Gabriel Nivasch, More on the Sprague-Grundy function for Wythoff's game, pages 377-410 in "Games of No Chance 3", MSRI Publications Volume 56, 2009.
Jeffrey Shallit, Automaton for A002251
Jeffrey Shallit, Proving properties of some greedily-defined integer recurrences via automata theory, arXiv:2308.06544 [cs.DM], 2023.
R. Silber, Wythoff's Nim and Fibonacci Representations, Fibonacci Quarterly #14 (1977), pp. 85-88.
N. J. A. Sloane, Scatterplot of first 100 terms [The points are symmetrically placed about the diagonal, although that is hard to see here because the scales on the axes are different]
Index entries for sequences that are permutations of the natural numbers

The sequence maps between A000201 and A001950, in that a(A000201(n)) = A001950(n), a(A001950(n)) = A000201(n).
Row 0 of A018219.
Cf. A073869, A342297.

With[{n = 42}, {0}~Join~Take[Values@ #, LengthWhile[#, # == 1 &] &@ Differences@ Keys@ #] &@ Sort@ Flatten@ Map[{#1 -> #2, #2 -> #1} & @@ # &, Transpose@ {Array[Floor[# GoldenRatio] &, n], Array[Floor[# GoldenRatio^2] &, n]}]] (* Michael De Vlieger, Nov 14 2017 *)

A002251_upto(N,c=0,A=Vec(0,N))={for(n=1,N, A[n]||(#AA002251[1]=2, a(0)=0 is not included. - M. F. Hasler, Nov 27 2019, replacing earlier code from Sep 17 2014

a(n) = A019444(n+1) - 1.

Edited by Christian G. Bower, Oct 29 2002

A367068 a(n) = ((Sum_{i=1..n} A340510(i))-1)/(n+1).

Original entry on oeis.org

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

Offset: 1

Zenan Sabanac, Dec 17 2023

nonn

For a positive integer k define the Avdispahić-Zejnulahi sequence AZ(k) by b(1)=k, and thereafter b(n) is the least positive integer not yet in the sequence such that Sum_{i=1..n} b(i) == k (mod n+k).
Define the Avdispahić-Zejnulahi means sequence AZM(k) by a(n) = ((Sum_{i=1..n} b(i))-k)/(n+k). This is the AZM(1) sequence.
Is this a duplicate of A005379? For n<=1300 at least a(n)=A005379(n). - R. J. Mathar, Jan 30 2024

Muharem Avdispahić and Faruk Zejnulahi, An integer sequence with a divisibility property, Fibonacci Quarterly, Vol. 58:4 (2020), 321-333.

Cf. A340510.

Cf. A073869 (AZM(0)), A367066 (AZM(2)).

A367068 := proc(n)
    add(A340510(i),i=1..n)-1 ;
    %/(n+1) ;
end proc:
seq(A367068(n),n=1..50) ; # R. J. Mathar, Jan 30 2024

zlist = {-1, 1, 3};
mlist = {-1, 0, 1};
For[n = 3, n <= 101, n++,
  If[MemberQ[zlist, mlist[[n]]], AppendTo[mlist, mlist[[n]] + 1];
    AppendTo[zlist, mlist[[n + 1]] + n];, AppendTo[mlist, mlist[[n]]];
     AppendTo[zlist, mlist[[n + 1]]];];];
mlist = Drop[mlist, 1]; mlist

z_list=[-1, 1, 3]
m_list=[-1, 0, 1]
n=2
for n in range(2, 100):
    if m_list[n] in z_list:
        m_list.append(m_list[n] + 1)
        z_list.append(m_list[n+1] + n+1)
    else:
        m_list.append(m_list[n])
        z_list.append(m_list[n+1])
print(m_list[1:])

For n>2, a(n) = a(n-1) if a(n-1) <> A340510(k) (for k=1..n-1) and a(n) = a(n-1)+1=A340510(n)-n otherwise. (See Proposition 3.1. of Avdispahić and Zejnulahi in the link above).

cp's OEIS Frontend

A090909 Terms a(k) of A073869 for which a(k-1) = a(k), and a(k) and a(k+1) are distinct.

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A090908 Terms a(k) of A073869 for which a(k) is distinct from a(k-1).

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A090638 Duplicate of A073869.

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A005206 Hofstadter G-sequence: a(0) = 0; a(n) = n - a(a(n-1)) for n > 0.

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A002251 Start with the nonnegative integers; then swap L(k) and U(k) for all k >= 1, where L = A000201, U = A001950 (lower and upper Wythoff sequences).

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A367068 a(n) = ((Sum_{i=1..n} A340510(i))-1)/(n+1).

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Python

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A367069 a(n) = ((Sum_{i=1..n} A367067(i))-3)/(n+3).

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