A191107 -id:A191107 - cp's OEIS frontend

A003278 Szekeres's sequence: a(n)-1 in ternary = n-1 in binary; also: a(1) = 1, a(2) = 2, and thereafter a(n) is smallest number k which avoids any 3-term arithmetic progression in a(1), a(2), ..., a(n-1), k.

1, 2, 4, 5, 10, 11, 13, 14, 28, 29, 31, 32, 37, 38, 40, 41, 82, 83, 85, 86, 91, 92, 94, 95, 109, 110, 112, 113, 118, 119, 121, 122, 244, 245, 247, 248, 253, 254, 256, 257, 271, 272, 274, 275, 280, 281, 283, 284, 325, 326, 328, 329, 334, 335, 337, 338, 352, 353

Offset: 1

N. J. A. Sloane and Richard Stanley

That is, there are no three elements A, B and C such that B - A = C - B.
Positions of 1's in Richard Stanley's Forest Fire sequence A309890. - N. J. A. Sloane, Dec 01 2019
Subtracting 1 from each term gives A005836 (ternary representation contains no 2's). - N. J. A. Sloane, Dec 01 2019
Difference sequence related to Gray code bit sequence (A001511). The difference patterns follows a similar repeating pattern (ABACABADABACABAE...), but each new value is the sum of the previous values, rather than simply 1 more than the maximum of the previous values. - Hal Burch (hburch(AT)cs.cmu.edu), Jan 12 2004
Sums of distinct powers of 3, translated by 1.
Positions of 0 in A189820; complement of A189822. - Clark Kimberling, May 26 2011
Also, Stanley sequence S(1): see OEIS Index under Stanley sequences (link below). - M. F. Hasler, Jan 18 2016
Named after the Hungarian-Australian mathematician George Szekeres (1911-2005). - Amiram Eldar, May 07 2021
If A_n=(a(1),a(2),...,a(2^n)), then A_(n+1)=(A_n,A_n+3^n). - Arie Bos, Jul 24 2022

			G.f. = x + 2*x^2 + 4*x^3 + 5*x^4 + 10*x^5 + 11*x^6 + 13*x^7 + 14*x^8 + 28*x^9 + ...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 164.
Richard K. Guy, Unsolved Problems in Number Theory, E10.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David W. Wilson, Table of n, a(n) for n = 1..10000 [a(1..1024) from T. D. Noe]
Jean-Paul Allouche and Jeffrey Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
Jean-Paul Allouche and Jeffrey Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
Paul Erdős and Paul Turan, On some sequences of integers, J. London Math. Soc., 11 (1936), 261-264.
Joseph Gerver, James Propp and Jamie Simpson, Greedily partitioning the natural numbers into sets free of arithmetic progressions Proc. Amer. Math. Soc. 102 (1988), no. 3, 765-772.
Fanel Iacobescu, Smarandache Partition Type and Other Sequences, Bull. Pure Appl. Sci. 16E, 237-240, 1997.
Henry Ibstedt, A Few Smarandache Sequences, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 170-183.
Gabor Korvin, Short note: Every large set of integers contains a three term arithmetic progression arXiv 1404.1557 [math.NT], Apr 6 2014.
Leo Moser, An Introduction to the Theory of Numbers, The Trillia Group, 2011 (written in 1957). See pp. 61-62.
James Propp and N. J. A. Sloane, Email, March 1994
Florentin Smarandache, Sequences of Numbers Involved in Unsolved Problems.
R. P. Stanley, Letter to N. J. A. Sloane, c. 1991
Eric Weisstein's World of Mathematics, Smarandache Sequences.
Index entries for sequences related to binary expansion of n
Index entries related to non-averaging sequences
Index entries related to Stanley sequences

Equals 1 + A005836. Cf. A001511, A098871.
Row 0 of array in A093682.
Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
3-term AP: A005836 (>=0), A003278 (>0);
4-term AP: A005839 (>=0), A005837 (>0);
5-term AP: A020654 (>=0), A020655 (>0);
6-term AP: A020656 (>=0), A005838 (>0);
7-term AP: A020657 (>=0), A020658 (>0);
8-term AP: A020659 (>=0), A020660 (>0);
9-term AP: A020661 (>=0), A020662 (>0);
10-term AP: A020663 (>=0), A020664 (>0).
Cf. A003002, A229037 (the Forest Fire sequence), A309890 (Stanley's version).
Similar formula:
If A_n=(a(1),a(2),...,a(2^n)), then A_(n+1)=(A_n,A_n+4^n) produces A098871;
If A_n=(a(1),a(2),...,a(2^n)), then A_(n+1)=(A_n,A_n+2*3^n) produces  A191106.

function a(n)
    return 1 + parse(Int, bitstring(n-1), base=3)
end # Gabriel F. Lipnik, Apr 16 2021

a:= proc(n) local m, r, b; m, r, b:= n-1, 1, 1;
      while m>0 do r:= r+b*irem(m, 2, 'm'); b:= b*3 od; r
    end:
seq(a(n), n=1..100); # Alois P. Heinz, Aug 17 2013

Take[ Sort[ Plus @@@ Subsets[ Table[3^n, {n, 0, 6}]]] + 1, 58] (* Robert G. Wilson v, Oct 23 2004 *)
a[1] = 0; h = 180;
Table[a[3 k - 2] = a[k], {k, 1, h}];
Table[a[3 k - 1] = a[k], {k, 1, h}];
Table[a[3 k] = 1, {k, 1, h}];
Table[a[n], {n, 1, h}]   (* A189820 *)
Flatten[Position[%, 0]]  (* A003278 *)
Flatten[Position[%%, 1]] (* A189822 *)
(* A003278 from A189820, from Clark Kimberling, May 26 2011 *)
Table[FromDigits[IntegerDigits[n, 2], 3] + 1, {n, 0, 57}] (* Amit Munje, Jun 03 2018 *)

a(n)=1+sum(i=1,n-1,(1+3^valuation(i,2))/2) \\ Ralf Stephan, Jan 21 2014

$nxt = 1; @list = (); for ($cnt = 0; $cnt < 1500; $cnt++) { while (exists $legal{$nxt}) { $nxt++; } print "$nxt "; last if ($nxt >= 1000000); for ($i = 0; $i <= $#list; $i++) { $t = 2*$nxt - $list[$i]; $legal{$t} = -1; } $cnt++; push @list, $nxt; $nxt++; } # Hal Burch

def A003278(n):
    return int(format(n-1,'b'),3)+1 # Chai Wah Wu, Jan 04 2015

a(2*k + 2) = a(2*k + 1) + 1, a(2^k + 1) = 2*a(2^k).
a(n) = b(n+1) with b(0) = 1, b(2*n) = 3*b(n)-2, b(2*n+1) = 3*b(n)-1. - Ralf Stephan, Aug 23 2003
G.f.: x/(1-x)^2 + x * Sum_{k>=1} 3^(k-1)*x^(2^k)/((1-x^(2^k))*(1-x)). - Ralf Stephan, Sep 10 2003, corrected by Robert Israel, May 25 2011
Conjecture: a(n) = (A191107(n) + 2)/3 = (A055246(n) + 5)/6. - L. Edson Jeffery, Nov 26 2015
a(n) mod 2 = A010059(n). - Arie Bos, Aug 13 2022

A191106 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-2 and 3x are in a.

Original entry on oeis.org

1, 3, 7, 9, 19, 21, 25, 27, 55, 57, 61, 63, 73, 75, 79, 81, 163, 165, 169, 171, 181, 183, 187, 189, 217, 219, 223, 225, 235, 237, 241, 243, 487, 489, 493, 495, 505, 507, 511, 513, 541, 543, 547, 549, 559, 561, 565, 567, 649, 651, 655, 657, 667, 669, 673, 675, 703, 705, 709, 711, 721, 723, 727, 729, 1459, 1461, 1465, 1467, 1477

Offset: 1

Clark Kimberling, May 26 2011

Related sequences for various choices of i and k as defined in A190803:
  A003278:  (i,k) = (-2,-1)
  A191106:  (i,k) = (-2, 0)
  A191107:  (i,k) = (-2, 1)
  A191108:  (i,k) = (-2, 2)
  A153775:  (i,k) = (-1, 0)
  A147991:  (i,k) = (-1, 1)
  A191109:  (i,k) = (-1, 2)
  A005836:  (i,k) = ( 0, 1)
  A191110:  (i,k) = ( 0, 2)
  A132140:  (i,k) = ( 1, 2)
For a=A191106, we have closure properties: the integers in (2+a)/3 comprise a; the integers in a/3 comprise a.
For k >= 1, m = a(i), 1 <= i <= 2^k seems to be m such that m/(3^k+1) is in the Cantor set (except that m = 0 and m = 3^k+1 do not appear). For k >= 2,  m = (a(i)-1)/2, 1 <= i <= 2^k seems to be m such that m/((3^k-1)/2) is in the Cantor set. - Peter Munn, Jul 06 2019
Every even number is the sum of two (possibly equal) terms. More specifically: terms a(1) through a(2^n) = 3^n sum to even numbers 2 times 1 through 3^n. Every even number is infinitely often the difference of two terms. Since the sequence is equal to 2*A005836(n) + 1, these properties follow immediately from similar properties of A005836 for every number. - Aad Thoen, Feb 17 2022
if A_n=(a(1),a(2),...,a(2^n)), then A_(n+1)=(A_n,A_n+2*3^n), similar to A003278. - Arie Bos, Jul 26 2022

			1 -> 3 -> 7,9 -> 19,21,25,27 -> ...

Ivan Neretin, Table of n, a(n) for n = 1..10000
David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, 10 (2007) 1-13.
D. Jordan and R. Schayer Rational points on the Cantor middle thirds set, Penn State, REU 2003.
Eric Weisstein's World of Mathematics, Cantor Set

Cf. A005823, A005836, A054591, A088917 (characteristic function), A173934, A190803, A191108.
Partial sums of A061393.
Similar formula as A003278, A_(n+1)=(A_n,A_n+2*3^n).

h = 3; i = -2; j = 3; k = 0; f = 1; g = 9;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A191106; regarding g, see note at A190803 *)
b = (a + 2)/3; c = a/3; r = Range[1, 900];
d = Intersection[b, r](* illustrates closure property *)
e = Intersection[c, r](* illustrates closure property *)
2 FromDigits[#, 3]&/@Tuples[{0, 1}, 7] + 1 (* Vincenzo Librandi, Jul 10 2019 *)

a(n) = 2*A005836(n) + 1. - Charles R Greathouse IV, Sep 06 2011
a(n) = A005823(n) + 1. - Vladimir Shevelev, Dec 17 2012
a(n) = (A191108(n) + 1)/2. - Peter Munn, Jul 09 2019

A055246 At step number k >= 1 the 2^(k-1) open intervals that are erased from [0,1] in the Cantor middle-third set construction are I(k,n) = (a(n)/3^k, (1+a(n))/3^k), n=1..2^(k-1).

Original entry on oeis.org

1, 7, 19, 25, 55, 61, 73, 79, 163, 169, 181, 187, 217, 223, 235, 241, 487, 493, 505, 511, 541, 547, 559, 565, 649, 655, 667, 673, 703, 709, 721, 727, 1459, 1465, 1477, 1483, 1513, 1519, 1531, 1537, 1621, 1627, 1639, 1645, 1675, 1681, 1693, 1699

Offset: 1

Wolfdieter Lang, May 23 2000

Related to A005836. Gives boundaries of open intervals that have to be erased in the Cantor middle-third set construction.
Let g(n) = Sum_{i=0..n} (i*binomial(n+i,i)^3*binomial(n,i)^2) = A112035(n). Let b = {m>0 : g(m) != 0 (mod 3)}. Then b(n) = a(n). - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 08 2004
Conjecture: Similarly to A191107, this increasing sequence is generated by the rules: a(1) = 1, and if x is in the sequence, then 3*x-2 and 3*x+4 are also in the sequence. - L. Edson Jeffery, Nov 17 2015

			k=1: (1/3, 2/3);
k=2: (1/9, 2/9), (7/9, 8/9);
k=3: (1/27, 2/27), (7/27, 8/27), (19/27, 20/27), (25/27, 26/27); ...

Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Index entries for 3-automatic sequences.

Cf. A003278, A005836, A005823, A055247, A112035.

Cf. A191107.

(* (Conjectured) Choose rows large enough to guarantee that all terms < max are generated. *)
rows = 1000; max = 10^4; a[1] = {1}; i = 1; Do[a[n_] = {}; Do[If[1 < 3*a[n - 1][[k]] - 2 < max, AppendTo[a[n], 3*a[n - 1][[k]] - 2], Break]; If[3*a[n - 1][[k]] + 4 < max, AppendTo[a[n], 3*a[n - 1][[k]] + 4], Break], {k, Length[a[n - 1]]}]; If[a[n] == {}, Break, i++], {n, 2, 1000}]; a055246 = Take[Flatten[Table[a[n], {n, i}]], 48] (* L. Edson Jeffery, Nov 17 2015 *)
Join[{1}, 1 + 6 Accumulate[Table[(3^IntegerExponent[n, 2] + 1)/2, {n, 60}]]] (* Vincenzo Librandi, Nov 26 2015 *)

g(n)=sum(i=0,n,i*binomial(n+i,i)^3*binomial(n,i)^2);
for (i=1,2000,if(Mod(g(i),3)<>0,print1(i,",")))

a(n) = fromdigits(binary(n-1),3)*6 + 1; \\ Kevin Ryde, Apr 23 2021

def A055246(n): return int(bin(n-1)[2:],3)*6|1 # Chai Wah Wu, Jun 26 2025

a(n) = 1+6*A005836(n), n >= 1.
a(n) = 1+3*A005823(n), n >= 1.
a(n+1) = A074938(n) + A074939(n); A074938: odd numbers in A005836, A074939: even numbers in A005836. - Philippe Deléham, Jul 10 2005
Conjecture: a(n) = 2*A191107(n) - 1 = 6*A003278(n) - 5 = (a((2*n-1)*2^(k-1))+2)/3^k, k>0. - L. Edson Jeffery, Nov 25 2015

Edited by N. J. A. Sloane, Nov 20 2015: used first comment to give more precise definition, and edited a comment at the suggestion of L. Edson Jeffery.

A094453 Numbers k such that binomial(2*k, k)/(k+2) is not an integer.

Original entry on oeis.org

1, 2, 4, 6, 7, 10, 13, 14, 25, 28, 30, 31, 34, 37, 40, 62, 79, 82, 85, 88, 91, 94, 106, 109, 112, 115, 118, 121, 126, 241, 244, 247, 250, 253, 254, 256, 268, 271, 274, 277, 280, 283, 322, 325, 328, 331, 334, 337, 349, 352, 355, 358, 361, 364, 510, 727, 730, 733

Offset: 1

Robert G. Wilson v, May 11 2004

A191107 is a subsequence as the relevant terms of A000984 are not divisible by 3 (see the comments in A005836 and A191107). - Peter Munn, Aug 14 2023

Numbers k such that either k + 2 is a power of 2, or k + 2 is divisible by 3 and none of the base-3 digits of k + 2 are 2 except possibly the second-last. See link for proof. Thus the sequence is the union of the positive terms of A00984 and of 9*k-2, 9*k + 1 and 9*k + 4 for k in A005836. - Robert Israel, Nov 17 2024

Robert Israel, Table of n, a(n) for n = 1..10000
Robert Israel, Characterization of A094453

Cf. A000108, A000984, A005836, A094575, A094576, A131381, A191107.

filter:= proc(n) local r,L;
  r:= n+2;
  if r = 2^padic:-ordp(r,2) then true
  else
    if r mod 3 <> 0 then false
    else
      L:= convert(r,base,3);
      not member(2,L[3..-1])
  fi fi
end proc:select(filter, [$1..1000]); # Robert Israel, Nov 17 2024

Select[ Range[735], Mod[Binomial[2#, # ], (# + 2)] != 0 &]

A265161 Array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = (3/2)*(3^k - 1) + A265159(n,k), n,k >= 1.

Original entry on oeis.org

8, 35, 26, 89, 107, 80, 116, 269, 323, 242, 251, 350, 809, 971, 728, 278, 755, 1052, 2429, 2915, 2186, 332, 836, 2267, 3158, 7289, 8747, 6560, 359, 998, 2510, 6803, 9476, 21869, 26243, 19682, 737, 1079, 2996, 7532, 20411, 28430, 65609, 78731, 59048

Offset: 1

L. Edson Jeffery, Dec 03 2015

Conjecture 1: The array contains without duplication all possible "gap numbers" as defined in A265100.

			Array A begins:
.      8    26    80    242    728    2186    6560    19682    59048
.     35   107   323    971   2915    8747   26243    78731   236195
.     89   269   809   2429   7289   21869   65609   196829   590489
.    116   350  1052   3158   9476   28430   85292   255878   767636
.    251   755  2267   6803  20411   61235  183707   551123  1653371
.    278   836  2510   7532  22598   67796  203390   610172  1830518
.    332   998  2996   8990  26972   80918  242756   728270  2184812
.    359  1079  3239   9719  29159   87479  262439   787319  2361959
.    737  2213  6641  19925  59777  179333  538001  1614005  4842017

Cf. A191107, A265100, A265104, A265159.

(* Array: *)
a005836[1] := 0; a005836[n_] := If[OddQ[n], 3*a005836[Floor[(n + 1)/2]], a005836[n - 1] + 1]; a265159[n_, k_] := 5 + 9*a005836[2^(k - 1)*(2 n - 1)]; a265161[n_, k_] := (3/2)*(3^k - 1) + a265159[n, k]; Grid[Table[a265161[n, k], {n, 9}, {k, 9}]]
(* Array antidiagonal flattened: *)
a005836[1] := 0; a005836[n_] := If[OddQ[n], 3*a005836[Floor[(n + 1)/2]], a005836[n - 1] + 1]; a265159[n_, k_] := 5 + 9*a005836[2^(k - 1)*(2 n - 1)]; a265161[n_, k_] := (3/2)*(3^k - 1) + a265159[n, k]; Flatten[Table[a265161[n - k + 1, k], {n, 9}, {k, n}]]

Conjecture 2: A(n,k) = A191107(n)*3^k - 1.

A338632 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 3x/(A(x) - 5x/(A(x) - 7x/(A(x) - 9x/(A(x) - ...))))), a continued fraction relation.

Original entry on oeis.org

1, 1, 2, 14, 166, 2714, 55866, 1377942, 39493518, 1288115570, 47086272754, 1906554619166, 84711219819062, 4098314765667082, 214489189682087594, 12075596389435432230, 727783484288200558110, 46755528594469120151010, 3189788089674119448202722

Offset: 0

Paul D. Hanna, Nov 04 2020

nonn

			G.f. A(x) = 1 + x + 2*x^2 + 14*x^3 + 166*x^4 + 2714*x^5 + 55866*x^6 + 1377942*x^7 + 39493518*x^8 + 1288115570*x^9 + 47086272754*x^10 + ...
where
1 = A(x) - x/(A(x) - 3*x/(A(x) - 5*x/(A(x) - 7*x/(A(x) - 9*x/(A(x) - 11*x/(A(x) - 13*x/(A(x) - 15*x/(A(x) - 17*x/(A(x) - 19*x/(A(x) - ...)))))))))), a continued fraction relation.

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A000699, A158119, A338636.

Cf. A191107, A186776.

{a(n) = my(A=[1],CF=1); for(i=1,n, A=concat(A,0); for(i=1,#A, CF = Ser(A) - (2*(#A-i)+1)*x/CF ); A[#A] = -polcoeff(CF,#A-1) );A[n+1] }
for(n=0,20,print1(a(n),", "))

a(n) = 2 (mod 4) for n > 1 (conjecture).
For n > 0, a(n) = 1 (mod 3) iff n = A191107(k) for some k >= 1 (conjecture).
For n > 0, a(n) = 2 (mod 3) iff n = A186776(k) for some k >= 2 where A186776 is the Stanley sequence S(0,2) (conjecture).
a(n) ~ 2^(2*n) * n^(n - 1/2) / (sqrt(Pi) * exp(n + 1/2)). - Vaclav Kotesovec, Nov 12 2020

A338636 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 3^2x/(A(x) - 5^2x/(A(x) - 7^2x/(A(x) - 9^2x/(A(x) - ...))))), a continued fraction relation.

Original entry on oeis.org

1, 1, 8, 272, 19480, 2353568, 429016872, 110046546096, 37825128764472, 16793443888112960, 9358539226503013960, 6397425528561882140240, 5264539843826571207135320, 5134140710880677886077086432, 5855644914993764696284947092840

Offset: 0

Paul D. Hanna, Nov 04 2020

nonn

			G.f. A(x) = 1 + x + 8*x^2 + 272*x^3 + 19480*x^4 + 2353568*x^5 + 429016872*x^6 + 110046546096*x^7 + 37825128764472*x^8 + 16793443888112960*x^9 + ...
where
1 = A(x) - x/(A(x) - 3^2*x/(A(x) - 5^2*x/(A(x) - 7^2*x/(A(x) - 9^2*x/(A(x) - 11^2*x/(A(x) - 13^2*x/(A(x) - 15^2*x/(A(x) - 17^2*x/(A(x) - 19^2*x/(A(x) - ...)))))))))), a continued fraction relation.

Paul D. Hanna, Table of n, a(n) for n = 0..150

Cf. A338632, A158119.

Cf. A191107, A186776.

{a(n) = my(A=[1],CF=1); for(i=1,n, A=concat(A,0); for(i=1,#A, CF = Ser(A) - (2*(#A-i)+1)^2*x/CF ); A[#A] = -polcoeff(CF,#A-1) );A[n+1] }
for(n=0,20,print1(a(n),", "))

a(n) = 0 (mod 8) for n > 1 (conjecture).
For n > 0, a(n) = 1 (mod 3) iff n = A191107(k) for some k >= 1 (conjecture).
For n > 0, a(n) = 2 (mod 3) iff n = A186776(k) for some k >= 2 where A186776 is the Stanley sequence S(0,2) (conjecture).
a(n) ~ 2^(6*n + 1) * n^(2*n - 1/2) / (Pi^(2*n + 1/2) * exp(2*n)). - Vaclav Kotesovec, Nov 12 2020

A326604 G.f. A(x) satisfies x / Series_Reversion(xA(x)) = (2A(x) + 1+x)/3.

Original entry on oeis.org

1, 1, 3, 21, 231, 3333, 58167, 1175877, 26827623, 679078677, 18844334727, 568229240901, 18491559492999, 645850960844469, 24099045218945031, 956889503377128261, 40291822946545245927, 1793614919867776690389, 84177429562216608349959, 4154548653801498090246597, 215137302566817565660007367, 11664210072689092804458508533

Offset: 0

Paul D. Hanna, Sep 21 2019

nonn

a(k) = 6 (mod 9) when k = A191107(n) for n > 1 (conjecture).

a(n) = 3*A249933(n) for n > 1.

			G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 231*x^4 + 3333*x^5 + 58167*x^6 + 1175877*x^7 + 26827623*x^8 + 679078677*x^9 + 18844334727*x^10 +...
such that
x/Series_Reversion(x*A(x)) = (2*A(x) + 1+x)/3 = 1 + x + 2*x^2 + 14*x^3 + 154*x^4 + 2222*x^5 + 38778*x^6 + 783918*x^7 + 17885082*x^8 + 452719118*x^9 + ...
ITERATIONS OF x*A(x).
Let G(x) = x*A(x), then
A(x) = 1 + G(x)/3 + G(G(x))*2/3^2 + G(G(G(x)))*2^2/3^3 + G(G(G(G(x))))*2^3/3^4 + G(G(G(G(G(x)))))*2^4/3^5 +...
The table of coefficients in the iterations of x*A(x) begin:
[1, 1, 3, 21, 231, 3333, 58167, 1175877, 26827623, ...];
[1, 2, 8, 58, 630, 8958, 154530, 3096330, 70161318, ...];
[1, 3, 15, 117, 1285, 18167, 310735, 6177745, 139076385, ...];
[1, 4, 24, 204, 2308, 32800, 559124, 11053668, 247451528, ...];
[1, 5, 35, 325, 3835, 55365, 946623, 18671961, 416326935, ...];
[1, 6, 48, 486, 6026, 89158, 1539350, 30423134, 677231222, ...];
[1, 7, 63, 693, 9065, 138383, 2427943, 48304893, 1076756889, ...];
[1, 8, 80, 952, 13160, 208272, 3733608, 75127944, 1682704256, ...];
[1, 9, 99, 1269, 18543, 305205, 5614887, 114768093, 2592154167, ...]; ...
in which the following sum along column k equals a(k+1):
a(2) = 3 = 1/3 + 2*2/9 + 3*4/27 + 4*8/81 + 5*16/243 + 6*32/729 +...
a(3) = 21 = 3/3 + 8*2/9 + 15*4/27 + 24*8/81 + 35*16/243 + 48*32/729 + ...
a(4) = 231 = 21/3 + 58*2/9 + 117*4/27 + 204*8/81 + 325*16/243 + 486*32/729 +...
a(5) = 3333 = 231*2/3 + 630*2/9 + 1285*4/27 + 2308*8/81 + 3835*16/243 + 6026*32/729 +...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A120956, A249933.

nmax = 21; sol = {a[0] -> 1}; Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - (1 + 2 A[x A[x] + O[x]^(n+1)])/(3-x), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 03 2019 *)

/* Prints N terms using x/Series_Reversion(x*A(x)) = (2*A(x) + 1+x)/3 */
N = 30; {A=[1, 1]; for(i=1, N, A = concat(A, -3*Vec(x/serreverse(x*Ser(concat(A, 0))))[#A+1]); print1(i, ", ") ); A}

G.f. A(x) satisfies:
(1) A( 3*x/(2*A(x) + 1+x) ) = (2*A(x) + 1+x)/3.
(2) A(x) = (1 + 2*A(x*A(x))) / (3-x).
(3) A(x) = 1 + Sum_{n>=1} G^n(x) * 2^(n-1)/3^n where G(x) = x*A(x) and G^n(x) = G^{n-1}(x*A(x)) denotes iteration with G^0(x) = x.

cp's OEIS Frontend

A003278 Szekeres's sequence: a(n)-1 in ternary = n-1 in binary; also: a(1) = 1, a(2) = 2, and thereafter a(n) is smallest number k which avoids any 3-term arithmetic progression in a(1), a(2), ..., a(n-1), k.

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A191106 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-2 and 3x are in a.

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A055246 At step number k >= 1 the 2^(k-1) open intervals that are erased from [0,1] in the Cantor middle-third set construction are I(k,n) = (a(n)/3^k, (1+a(n))/3^k), n=1..2^(k-1).

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A094453 Numbers k such that binomial(2*k, k)/(k+2) is not an integer.

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A265161 Array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = (3/2)*(3^k - 1) + A265159(n,k), n,k >= 1.

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A338632 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 3*x/(A(x) - 5*x/(A(x) - 7*x/(A(x) - 9*x/(A(x) - ...))))), a continued fraction relation.

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A338636 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 3^2*x/(A(x) - 5^2*x/(A(x) - 7^2*x/(A(x) - 9^2*x/(A(x) - ...))))), a continued fraction relation.

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A338632 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 3x/(A(x) - 5x/(A(x) - 7x/(A(x) - 9x/(A(x) - ...))))), a continued fraction relation.

A338636 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 3^2x/(A(x) - 5^2x/(A(x) - 7^2x/(A(x) - 9^2x/(A(x) - ...))))), a continued fraction relation.

A326604 G.f. A(x) satisfies x / Series_Reversion(xA(x)) = (2A(x) + 1+x)/3.