A005229 -id:A005229 - cp's OEIS frontend

A147880 Expansion of Product_{k > 0} (1 + A005229(k)*x^k).

1, 1, 1, 3, 5, 8, 12, 21, 30, 50, 75, 110, 169, 249, 361, 539, 757, 1076, 1583, 2207, 3121, 4415, 6184, 8468, 11775, 16274, 22314, 30601, 41745, 56412, 77008, 103507, 138383, 186928, 249855, 333375, 443898, 588402, 778276, 1031126, 1356945, 1780645

Offset: 0

Roger L. Bagula, Nov 16 2008

nonn

			From _Petros Hadjicostas_, Apr 10 2020: (Start)
Let f(m) = A005229(m). Using the strict partitions of each n (see A000009), we get
a(1) = f(1) = 1,
a(2) = f(2) = 1,
a(3) = f(3) + f(1)*f(2) = 2 + 1*1 = 3,
a(4) = f(4) + f(1)*f(3) = 3 + 1*2 = 5,
a(5) = f(5) + f(1)*f(4) + f(2)*f(3) = 3 + 1*3 + 1*2 = 8,
a(6) = f(6) + f(1)*f(5) + f(2)*f(4) + f(1)*f(2)*f(3) = 4 + 1*3 + 1*3 + 1*1*2 = 12,
a(7) = f(7) + f(1)*f(6) + f(2)*f(5) + f(3)*f(4) + f(1)*f(2)*f(4) = 5 + 1*4 + 1*3 + 2*3 + 1*1*3 = 21. (End)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000009, A004001, A005229, A147559, A147654, A147655, A147665.

(*A005229*) f[n_Integer?Positive] := f[n] = f[ f[n - 2]] + f[n - f[n - 2]]; f[0] = 0; f[1] = f[2] = 1;
P[x_, n_] := P[x, n] = Product[1 + f[m] *x^m, {m, 0, n}];
Take[CoefficientList[P[x, 45], x], 45] (* Program simplified by Petros Hadjicostas, Apr 13 2020 *)

\\ here B(n) is A005229 as vector.
B(n)={my(a=vector(n, i, 1)); for(n=3, n, a[n] = a[a[n-2]] + a[n-a[n-2]]); a}
seq(n)={my(v=B(n)); Vec(prod(k=1, n, 1 + v[k]*x^k + O(x*x^n)))} \\ Andrew Howroyd, Apr 10 2020

G.f.: Product_{k > 0} (1 + A005229(k)*x^k).

a(n) = Sum_{(b_1,...,b_n)} f(1)^b_1 * f(2)^b_2 * ... * f(n)^b_n, where f(m) = A005229(m), and the sum is taken over all lists (b_1,...,b_n) with b_j in {0,1} and Sum_{j=1..n} j*b_j = n.

Various sections edited by Joerg Arndt and Petros Hadjicostas, Apr 10 2020

A051105 First differences of A005229.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A005229.

b[n_]:= b[n]= If[n<3, 1, b[b[n-2]] + b[n-b[n-2]]]; (* A005229 *)
A051105[n_]:= A051105[n] = b[n+1] -b[n];
Table[A051105[n], {n, 100}] (* G. C. Greubel, Mar 27 2022 *)

@CachedFunction
def b(n): # A005229
    if (n<3): return 1
    else: return b(b(n-2)) + b(n-b(n-2))
def A051105(n): return b(n+1) - b(n)
[A051105(n) for n in (1..100)] # G. C. Greubel, Mar 27 2022

a(n) = A005229(n+1) - A005229(n). - G. C. Greubel, Mar 28 2022

A087758 Index of first occurrence of n in A005229.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 10, 12, 13, 14, 16, 17, 19, 20, 21, 22, 24, 26, 27, 29, 31, 32, 33, 34, 36, 37, 39, 40, 41, 43, 46, 47, 48, 49, 50, 51, 53, 55, 56, 58, 59, 60, 61, 63, 64, 67, 69, 70, 71, 72, 73, 74, 75, 77, 80, 81, 82, 83, 85, 87, 88, 89, 90, 92, 94, 95, 98, 99, 100, 102

Offset: 1

Roger L. Bagula, Oct 02 2003

nonn

Cf. A005229.

{m=102;v5229=vector(m);v5229[1]=1;v5229[2]=1;for(k=3,m,v5229[k]=v5229[v5229[k-2]]+v5229[k-v5229[k-2]]); v=vector(m);for(j=1,m,if(v[v5229[j]]==0,v[v5229[j]]=j));n=0;while(v[n++ ]>0,print1(v[n],","))}

a(n) = least k such that A005229(k) = n.

Edited by N. J. A. Sloane, Apr 07 2006

A116591 a(n) = b(n+2) + b(n) with a(0) = 1, where b(n) = A005229(n) for n>2.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 10, 11, 13, 13, 15, 16, 18, 19, 21, 22, 23, 25, 26, 28, 30, 31, 33, 33, 35, 36, 37, 39, 39, 41, 42, 44, 46, 47, 49, 50, 51, 53, 54, 56, 57, 59, 59, 60, 61, 62, 64, 66, 68, 70, 71, 73, 73, 75, 76, 77, 79, 80, 82, 84, 85, 87, 88, 89, 90, 91, 91, 93, 94, 96, 98

Offset: 0

Roger L. Bagula, Mar 27 2006

nonn

A similar definition applied to the Fibonacci sequence (A000045) leads to the Lucas sequence (A000032).

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A000032, A005185, A005229.

b:=proc(n) option remember; if n<=2 then 1 else b(b(n-2))+b(n-b(n-2)): fi: end: seq(b(n),n=1..75): a[0]:=1: for n from 1 to 70 do a[n]:=b(n)+b(n+2) od: seq(a[n],n=0..70);

M[n_]:= M[n]= If[n<3, 1 -Boole[n==0], M[M[n-2]] + M[n -M[n-2]]];
L[n_]:= L[n]= If[n==1, 1, M[n-1] + M[n+1]];
Table[L[n], {n, 100}] (* modified by G. C. Greubel, Mar 28 2022 *)

@CachedFunction
def b(n): # A005229
    if (n<3): return 1
    else: return b(b(n-2)) + b(n-b(n-2))
def A116591(n): return b(n+2) +b(n) -bool(n==0)
[A116591(n) for n in (0..100)] # G. C. Greubel, Mar 28 2022

a(n) = A005229(n+2) + A005229(n) for n>=1.

Edited by N. J. A. Sloane, Apr 15 2006

A169638 Number of multiset permutations of the n initial elements of A005229 with additional element A005229(0)=1.

Original entry on oeis.org

1, 1, 1, 4, 20, 60, 420, 3360, 30240, 151200, 1663200, 9979200, 129729600, 1816214400, 27243216000, 217945728000, 3705077376000, 66691392768000, 633568231296000, 12671364625920000, 266098657144320000, 5854170457175040000, 134645920515025920000, 1615751046180311040000

Offset: 0

Roger L. Bagula, Apr 04 2010

nonn

Robert Israel, Table of n, a(n) for n = 0..429

Cf. A169637.

N:= 100: # to get a(0) to a(N)
A005229:= proc(n) option remember;
procname(procname(n-2))+procname(n-procname(n-2))
end proc:
A005229(1):= 1: A005229(2):= 1:
V:= Vector(N):
A[0]:= 1: V[1]:= 1:
for n from 1 to N do
  r:= A005229(n);
  V[r]:= V[r]+1;
  A[n]:= A[n-1]*(n+1)/V[r];
od:
seq(A[i],i=0..N); # Robert Israel, Dec 23 2014

Mallows[n_Integer?Positive] := Mallows[n] = Mallows[Mallows[n - 2]] + Mallows[ n - Mallows[n - 2]];
Mallows[0] = Mallows[1] = Mallows[2] = 1;
a[m_] := Length[Permutations[Table[Mallows[i], {i, 0, m}]]];
Table[a[m], {m, 0, 10}]
(* A much better way to compute the terms is to use the multinomials of the multiplicities of the terms of A005229! - Joerg Arndt, Dec 23 2014 *)

a(n) = number of permutations of the list b[0..n] where b(0)=0 and b(n) = A005229(n) for n>=1.

Edited and new name, Joerg Arndt, Dec 23 2014

a(11) to a(23) from Robert Israel, Dec 23 2014

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++)}

A152569 Terms in A005229 occurring exactly 3 times.

Original entry on oeis.org

30, 45, 54, 66, 78, 93, 97, 113, 118, 135, 142, 143, 163, 171, 195, 205, 211, 235, 247, 278, 297, 310, 339, 355, 379, 401, 429, 432, 455, 490, 510, 519, 524, 548, 555, 578, 618, 627, 640, 646, 654, 658, 668, 734, 746, 759, 790, 797, 834, 846, 852, 875, 890

Offset: 1

Zak Seidov, Dec 08 2008

nonn

A005229 a(1)=a(2)=1; for n>2, a(n)=a(a(n-2))+a(n-a(n-2)).

aa[1] = aa[2] = 1; aa[n_] := aa[n] = aa[aa[n - 2]] + aa[n - aa[n - 2]]; ta2000 = Table[aa[n], {n, 2000}]; re3 = Reap[Do[If[Count[ta2000, k] == 3, Sow[k]], {k, 1379}]][[2, 1]]

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]

A028310 Expansion of (1 - x + x^2) / (1 - x)^2 in powers of x.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 0

N. J. A. Sloane

1 followed by the natural numbers.
Molien series for ring of Hamming weight enumerators of self-dual codes (with respect to Euclidean inner product) of length n over GF(4).
Engel expansion of e (see A006784 for definition) [when offset by 1]. - Henry Bottomley, Dec 18 2000
Also the denominators of the series expansion of log(1+x). Numerators are A062157. - Robert G. Wilson v, Aug 14 2015
The right-shifted sequence (with a(0)=0) is an autosequence (of the first kind - see definition in links). - Jean-François Alcover, Mar 14 2017

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 8*x^8 + 9*x^9  + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
Daniel Birmajer, Juan B. Gil, Jordan O. Tirrell, and Michael D. Weiner, Pattern-avoiding stabilized-interval-free permutations, arXiv:2306.03155 [math.CO], 2023.
Olivia Nabawanda and Fanja Rakotondrajao, The sets of flattened partitions with forbidden patterns, arXiv:2011.07304 [math.CO], 2020.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Oeis Wiki, Autosequence
E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (2,-1).
Index entries for Molien series
Index entries for sequences related to Engel expansions

Cf. A000007, A000027, A000660 (boustrophedon transform).

Cf. A001477, A004001, A005229, A112934, A123110, A212393, A204503.

a028310 n = 0 ^ n + n
a028310_list = 1 : [1..]  -- Reinhard Zumkeller, Nov 06 2012

[n eq 0 select 1 else n: n in [0..75]]; // G. C. Greubel, Jan 05 2024

a:= n-> `if`(n=0, 1, n):
seq(a(n), n=0..60);

Denominator@ CoefficientList[Series[Log[1+x], {x,0,75}], x] (* or *)
CoefficientList[ Series[(1 -x +x^2)/(1-x)^2, {x,0,75}], x] (* Robert G. Wilson v, Aug 14 2015 *)
Join[{1}, Range[75]] (* G. C. Greubel, Jan 05 2024 *)
LinearRecurrence[{2,-1},{1,1,2},80] (* Harvey P. Dale, Jan 29 2025 *)

{a(n) = (n==0) + max(n, 0)} /* Michael Somos, Feb 02 2004 */

A028310(n)=n+!n  \\ M. F. Hasler, Jan 16 2012

def A028310(n): return n|bool(n)^1 # Chai Wah Wu, Jul 13 2023

[n + int(n==0) for n in range(76)] # G. C. Greubel, Jan 05 2024

Binomial transform is A005183. - Paul Barry, Jul 21 2003
G.f.: (1 - x + x^2) / (1 - x)^2 = (1 - x^6) /((1 - x) * (1 - x^2) * (1 - x^3)) = (1 + x^3) / ((1 - x) * (1 - x^2)). a(0) = 1, a(n) = n if n>0.
Euler transform of length 6 sequence [ 1, 1, 1, 0, 0, -1]. - Michael Somos Jul 30 2006
G.f.: 1 / (1 - x / (1 - x / (1 + x / (1 - x)))). - Michael Somos, Apr 05 2012
G.f. of A112934(x) = 1 / (1 - a(0)*x / (1 - a(1)*x / ...)). - Michael Somos, Apr 05 2012
a(n) = A000027(n) unless n=0.
a(n) = Sum_{k=0..n} A123110(n,k). - Philippe Deléham, Oct 06 2009
E.g.f: 1+x*exp(x). - Wolfdieter Lang, May 03 2010
a(n) = sqrt(floor[A204503(n+3)/9]). - M. F. Hasler, Jan 16 2012
E.g.f.: 1-x + x*E(0), where E(k) = 2 + x/(2*k+1 - x/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 24 2013
a(n) = A001477(n) + A000007(n). - Miko Labalan, Dec 12 2015 (See the first comment.)

cp's OEIS Frontend

A147880 Expansion of Product_{k > 0} (1 + A005229(k)*x^k).

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A051105 First differences of A005229.

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A087758 Index of first occurrence of n in A005229.

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A116591 a(n) = b(n+2) + b(n) with a(0) = 1, where b(n) = A005229(n) for n>2.

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A169638 Number of multiset permutations of the n initial elements of A005229 with additional element A005229(0)=1.

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A118179 Numbers which occur only once in A005229.

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A152569 Terms in A005229 occurring exactly 3 times.

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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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