A003410 -id:A003410 - cp's OEIS frontend

A038718 Number of permutations P of {1,2,...,n} such that P(1)=1 and |P^-1(i+1)-P^-1(i)| equals 1 or 2 for i=1,2,...,n-1.

1, 1, 2, 4, 6, 9, 14, 21, 31, 46, 68, 100, 147, 216, 317, 465, 682, 1000, 1466, 2149, 3150, 4617, 6767, 9918, 14536, 21304, 31223, 45760, 67065, 98289, 144050, 211116, 309406, 453457, 664574, 973981, 1427439, 2092014, 3065996, 4493436, 6585451

Offset: 1

John W. Layman, May 02 2000

This sequence is the number of digits of each term of A061583. - Dmitry Kamenetsky, Jan 17 2009

Michael De Vlieger, Table of n, a(n) for n = 1..6023
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-1).

Cf. A003274, A003410.

LinearRecurrence[{2,-1,1,-1},{1,1,2,4},50] (* or *) CoefficientList[ Series[(x^2-x+1)/(x^4-x^3+x^2-2x+1),{x,0,50}],x] (* Harvey P. Dale, Apr 24 2011 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,1,-1,2]^(n-1)*[1;1;2;4])[1,1] \\ Charles R Greathouse IV, Apr 07 2016

From Joseph Myers, Feb 03 2004: (Start)
G.f.: (1 -x +x^2)/(1-2*x+x^2-x^3+x^4).
a(n) = a(n-1) + a(n-3) + 1. (End)
a(n) = Sum_{i=1..n} A058278(i) = A097333(n) - 1. - R. J. Mathar, Oct 16 2010

More terms from Joseph Myers, Feb 03 2004

A097333 a(n) = Sum_{k=0..n} C(n-k, floor(k/2)).

Original entry on oeis.org

1, 2, 2, 3, 5, 7, 10, 15, 22, 32, 47, 69, 101, 148, 217, 318, 466, 683, 1001, 1467, 2150, 3151, 4618, 6768, 9919, 14537, 21305, 31224, 45761, 67066, 98290, 144051, 211117, 309407, 453458, 664575, 973982, 1427440, 2092015, 3065997, 4493437, 6585452

Offset: 0

Paul Barry, Aug 05 2004

Partial sums of A058278.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Kassie Archer and Aaron Geary, Powers of permutations that avoid chains of patterns, arXiv:2312.14351 [math.CO], 2023. See p. 15.
Engin Özkan, Bahar Kuloǧlu, and James Peters, k-Narayana sequence self-similarity, hal-03242990 [math.CO], 2021. See p. 12.
Index entries for linear recurrences with constant coefficients, signature (1,0,1).

Essentially the same as A003410 and A058278.

LinearRecurrence[{1, 0, 1}, {1, 2, 2}, 70] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2011*)

a(n) = sum(k=0, n, binomial(n-k, k\2)); \\ Michel Marcus, Mar 02 2022

G.f.: (1+x-x^2-x^3)/((1-x)*(1-x^2-x^3-x^4)) = (1+x)/(1-x-x^3);
a(n) = a(n-1) + a(n-3);
a(n) = a(n-1) + a(n-2) - a(n-5).
a(n) = A058278(n+3) = A000930(n-1)+A000930(n). - R. J. Mathar, Jul 07 2023
a(n) = A003410(n-1) for n >= 2. - Jianing Song, Aug 11 2023

A226130 Denominators of rational numbers as generated by the rules: 1 is in S, and if nonzero x is in S, then x+1 and -1/x are in S. (See Comments.)

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 4, 3, 2, 1, 1, 5, 4, 3, 2, 2, 3, 1, 6, 5, 4, 3, 3, 5, 2, 5, 3, 1, 7, 6, 5, 4, 4, 7, 3, 8, 5, 2, 7, 5, 3, 1, 1, 8, 7, 6, 5, 5, 9, 4, 11, 7, 3, 11, 8, 5, 2, 2, 9, 7, 5, 3, 3, 4, 1, 9, 8, 7, 6, 6, 11, 5, 14, 9, 4, 15, 11, 7, 3, 3

Offset: 1

Clark Kimberling, May 28 2013

Let S be the set of numbers defined by these rules:  1 is in S, and if nonzero x is in S, then x + 1 and -1/x are in S.  Then S is the set of all rational numbers, produced in generations as follows: g(1) = (1), g(2) = (2, -1), g(3) = (3, -1/2, 0), g(4) = (4, -1/3, 1/2), ... For n > 4, once g(n-1) = (c(1), ..., c(z)) is defined, g(n) is formed from the vector (c(1)+1, -1/c(1), c(2)+1, -1/c(2), ..., c(z)+1, -1/c(z)) by deleting previously generated elements.  Let S' denote the sequence formed by concatenating the generations.
A226130:  Denominators of terms of S'
A226131:  Numerators of terms of S'
A226136:  Positions of positive integers in S'
A226137:  Positions of integers in S'
The length of row n is given by A226275(n-1). - Peter Kagey, Jan 17 2022

			The denominators and numerators are read from the rationals in S':
  1/1, 2/1, -1/1, 3/1, -1/2, 0/1, 4/1, -1/3, 1/2, ...
Table begins:
  n |
  --+-----------------------------------------------
  1 | 1;
  2 | 1, 1;
  3 | 1, 2, 1;
  4 | 1, 3, 2;
  5 | 1, 4, 3, 2, 1;
  6 | 1, 5, 4, 3, 2, 2, 3;
  7 | 1, 6, 5, 4, 3, 3, 5, 2, 5, 3;
  8 | 1, 7, 6, 5, 4, 4, 7, 3, 8, 5, 2, 7, 5, 3, 1;

Clark Kimberling, Table of n, a(n) for n = 1..1000
Index entries for fraction trees

Cf. A226080 (rabbit ordering of positive rationals).
Cf. A226130, A226131, A226136, A226137, A226275.
Cf. A226247 (analogous with "0 is in S").

g[1] := {1}; z = 20; g[n_] := g[n] = DeleteCases[Flatten[Transpose[{# + 1, -1/#}]]&[DeleteCases[g[n - 1], 0]], Apply[Alternatives, Flatten[Map[g, Range[n - 1]]]]]; Flatten[Map[g, Range[7]]]  (* ordered rationals *)
Map[g, Range[z]]; Table[Length[g[i]], {i, 1, z}] (* cf. A003410 *)
f = Flatten[Map[g, Range[z]]];
Take[Denominator[f], 100] (* A226130 *)
Take[Numerator[f], 100]   (* A226131 *)
p1 = Flatten[Table[Position[f, n], {n, 1, z}]] (* A226136 *)
p2 = Flatten[Table[Position[f, -n], {n, 0, z}]];
Union[p1, p2]  (* A226137 *)  (* Peter J. C. Moses, May 26 2013 *)

from fractions import Fraction
from itertools import count, islice
def agen():
    rats = [Fraction(1, 1)]
    seen = {Fraction(1, 1)}
    for n in count(1):
        yield from [r.denominator for r in rats]
        newrats = []
        for r in rats:
            f = 1+r
            if f not in seen:
                newrats.append(1+r)
                seen.add(f)
            if r != 0:
                g = -1/r
                if g not in seen:
                    newrats.append(-1/r)
                    seen.add(g)
        rats = newrats
print(list(islice(agen(), 84))) # Michael S. Branicky, Jan 17 2022

A058278 Expansion of (1 - x^2)/(1 - x - x^3).

Original entry on oeis.org

1, 1, 0, 1, 2, 2, 3, 5, 7, 10, 15, 22, 32, 47, 69, 101, 148, 217, 318, 466, 683, 1001, 1467, 2150, 3151, 4618, 6768, 9919, 14537, 21305, 31224, 45761, 67066, 98290, 144051, 211117, 309407, 453458, 664575, 973982, 1427440, 2092015, 3065997, 4493437

Offset: 0

Robert G. Wilson v, Dec 06 2000

G. C. Greubel, Table of n, a(n) for n = 0..1000
Engin Özkan, Bahar Kuloǧlu, and James Peters, k-Narayana sequence self-similarity, hal-03242990 [math.CO], 2021. See p. 12.
Index entries for linear recurrences with constant coefficients, signature (1,0,1).

Essentially the same as A003410 and A097333.

Cf. A000930.

CoefficientList[Series[(1 - x^2)/(1 - x - x^3), {x, 0, 50}], x] (* Vladimir Joseph Stephan Orlovsky, Dec 28 2010 *)
LinearRecurrence[{1,0,1},{1,1,0},50] (* Harvey P. Dale, Apr 03 2015 *)

Vec((1-x^2)/(1-x-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

G.f.: (1 - x^2)/(1 - x - x^3).
a(n+3) = Sum_{k=0..n} binomial(n-k, floor(k/2)). - Paul Barry, Jul 06 2004
a(n) = a(n-3) + a(n-1). - Graeme McRae, Apr 26 2010
From Wolfdieter Lang, Apr 21 2015 : (Start)
a(n) = A097333(n-3), n >= 3.
a(n) = A000930(n) - A000930(n-2), n >= 2. (End)
a(n) = A003410(n-4) for n >= 5. - Jianing Song, Aug 11 2023

Edited: Offset corrected to 0. The formula by P. Barry corrected. Old formulas adapted to new offset. - Wolfdieter Lang, Apr 21 2015

A226131 Numerators of rational numbers as generated by the rules: 1 is in S, and if nonzero x is in S, then x+1 and -1/x are in S. (See Comments.)

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 28 2013

Let S be the set of numbers defined by these rules:  1 is in S, and if nonzero x is in S, then x + 1 and -1/x are in S.  Then S is the set of all rational numbers, produced in generations as follows: g(1) = (1), g(2) = (2, -1), g(3) = (3, -1/2, 0), g(4) = (4, -1/3, 1/2), ... For n > 4, once g(n-1) = (c(1), ..., c(z)) is defined, g(n) is formed from the vector (c(1)+1, -1/c(1), c(2)+1, -1/c(2), ..., c(z)+1, -1/c(z)) by deleting previously generated elements.
Let S' denote the sequence formed by concatenating the generations.
  A226130:  Denominators of terms of S'
  A226131:  Numerators of terms of S'
  A226136:  Positions of positive integers in S'
  A226137:  Positions of integers in S'

			Rationals in S': 1/1, 2/1, -1/1, 3/1, -1/2, 0/1, 4/1, -1/3, 1/2, ...

Clark Kimberling, Table of n, a(n) for n = 1..1000
Peter Kagey, Illustration of the first seven generations.
Index entries for fraction trees

Cf. A226080 (rabbit ordering of positive rationals), A226247.

Cf. A226130, A226136, A226137.

g[1] := {1}; z = 20; g[n_] := g[n] = DeleteCases[Flatten[Transpose[{# + 1, -1/#}]]&[DeleteCases[g[n - 1], 0]], Apply[Alternatives, Flatten[Map[g, Range[n - 1]]]]]; Flatten[Map[g, Range[7]]]  (* ordered rationals *)
Map[g, Range[z]]; Table[Length[g[i]], {i, 1, z}] (* cf A003410 *)
f = Flatten[Map[g, Range[z]]];
Take[Denominator[f], 100] (* A226130 *)
Take[Numerator[f], 100]    (* A226131 *)
p1 = Flatten[Table[Position[f, n], {n, 1, z}]] (* A226136 *)
p2 = Flatten[Table[Position[f, -n], {n, 0, z}]];
Union[p1, p2]  (* A226137 *) (* Peter J. C. Moses, May 26 2013 *)

A101399 a(0) = 1, a(1) = 2, a(2) = 5; for n >= 3, a(n) = a(n-1) + 2*a(n-2) + a(n-3).

Original entry on oeis.org

1, 2, 5, 10, 22, 47, 101, 217, 466, 1001, 2150, 4618, 9919, 21305, 45761, 98290, 211117, 453458, 973982, 2092015, 4493437, 9651449, 20730338, 44526673, 95638798, 205422482, 441226751, 947710513, 2035586497, 4372234274

Offset: 0

Jeroen F.J. Laros, Jan 15 2005

Lengths of successive words (starting with a) under the substitution: {a -> ab, b -> aac, c -> a}.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Dun Qiu, Classical pattern distributions in Sn(132) and Sn(123), arXiv:1810.10099 [math.CO], 2018. See p. 11.
Index entries for linear recurrences with constant coefficients, signature (1, 2, 1).

Pairwise sums of A078007. Bisection of A003410 and A058278.

a:=[1,2,5];; for n in [4..35] do a[n]:=a[n-1]+2*a[n-2]+a[n-3]; od; a; # Muniru A Asiru, Apr 03 2018

I:=[1,2,5]; [n le 3 select I[n] else Self(n-1) + 2*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Apr 03 2018

m:=25; R:=PowerSeriesRing( Integers(), m); Coefficients(R!((1+x+x^2)/(1-x-2*x^2-x^3))); // G. C. Greubel, Apr 03 2018

a[0] = 1; a[1] = 2; a[2] = 5; a[n_] := a[n] = a[n - 1] + 2a[n - 2] + a[n - 3]; Table[ a[n], {n, 0, 30}] (* Robert G. Wilson v, Jan 15 2005 *)
LinearRecurrence[{1,2,1},{1,2,5},30] (* Harvey P. Dale, Aug 29 2012 *)
CoefficientList[Series[(1+x+x^2)/(1-x-2*x^2-x^3), {x, 0, 50}], x] (* G. C. Greubel, Apr 03 2018 *)

x='x+O('x^30); Vec((1+x+x^2)/(1-x-2*x^2-x^3)) \\ G. C. Greubel, Apr 03 2018

G.f.: (1+x+x^2)/(1-x-2*x^2-x^3). - G. C. Greubel, Apr 03 2018

More terms from Robert G. Wilson v and Lior Manor, Jan 15 2005

A226136 Positions of the positive integers in the ordering of rational numbers as generated by the rules: 1 is in S, and if nonzero x is in S, then x+1 and -1/x are in S. (See Comments.)

Original entry on oeis.org

1, 2, 4, 7, 10, 15, 22, 32, 47, 69, 101, 148, 217, 318, 466, 683, 1001, 1467, 2150, 3151, 4618, 6768, 9919, 14537, 21305, 31224, 45761, 67066, 98290, 144051, 211117, 309407, 453458, 664575, 973982

Offset: 1

Clark Kimberling, May 28 2013

nonn

Let S be the set of numbers defined by these rules:  1 is in S, and if nonzero x is in S, then x + 1 and -1/x are in S.  Then S is the set of all rational numbers, produced in generations as follows: g(1) = (1), g(2) = (2, -1), g(3) = (3, -1/2, 0), g(4) = (4, -1/3, 1/2), ... For n > 4, once g(n-1) = (c(1), ..., c(z)) is defined, g(n) is formed from the vector (c(1)+1, -1/c(1), c(2)+1, -1/c(2), ..., c(z)+1, -1/c(z)) by deleting previously generated elements.  Let S' denote the sequence formed by concatenating the generations.
A226130:  Denominators of terms of S'
A226131:  Numerators of terms of S'
A226136:  Positions of positive integers in S'
A226137:  Positions of integers in S'

			S' = (1/1, 2/1, -1/1, 3/1, -1/2, 0/1, 4/1, -1/3, 1/2, ...), with positive integers appearing in positions 1,2,4,7,...

Clark Kimberling, Table of n, a(n) for n = 1..35

Cf. A226080 (rabbit ordering of positive rationals).

g[1] := {1}; z = 20; g[n_] := g[n] = DeleteCases[Flatten[Transpose[{# + 1, -1/#}]]&[DeleteCases[g[n - 1], 0]], Apply[Alternatives, Flatten[Map[g, Range[n - 1]]]]]; Flatten[Map[g, Range[7]]]  (* ordered rationals *)
Map[g, Range[z]]; Table[Length[g[i]], {i, 1, z}] (* cf A003410 *)
f = Flatten[Map[g, Range[z]]];
Take[Denominator[f], 100] (* A226130 *)
Take[Numerator[f], 100]   (* A226131 *)
p1 = Flatten[Table[Position[f, n], {n, 1, z}]] (* A226136 *)
p2 = Flatten[Table[Position[f, -n], {n, 0, z}]];
Union[p1, p2]  (* A226137 *) (* Peter J. C. Moses, May 26 2013 *)

Conjecture: a(n) = a(n-1)+a(n-3) for n>6. G.f.: -x*(x+1) * (x^2+1)^2 / (x^3+x-1). - Colin Barker, Jul 03 2013

A226137 Positions of the integers in the ordering of rational numbers as generated by the rules: 1 is in S, and if nonzero x is in S, then x+1 and -1/x are in S. (See Comments.)

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 10, 14, 15, 22, 32, 46, 47, 69, 101, 147, 148, 217, 318, 465, 466, 683, 1001, 1466, 1467, 2150, 3151, 4617, 4618, 6768, 9919, 14536, 14537, 21305, 31224, 45760, 45761, 67066, 98290, 144050, 144051, 211117, 309407, 453457, 453458

Offset: 1

Clark Kimberling, May 28 2013

nonn

Let S be the set of numbers defined by these rules:  1 is in S, and if nonzero x is in S, then x + 1 and -1/x are in S.  Then S is the set of all rational numbers, produced in generations as follows: g(1) = (1), g(2) = (2, -1), g(3) = (3, -1/2, 0), g(4) = (4, -1/3, 1/2), ... For n > 4, once g(n-1) = (c(1), ..., c(z)) is defined, g(n) is formed from the vector (c(1)+1, -1/c(1), c(2)+1, -1/c(2), ..., c(z)+1, -1/c(z)) by deleting previously generated elements.  Let S' denote the sequence formed by concatenating the generations.
A226130:  Denominators of terms of S'
A226131:  Numerators of terms of S'
A226136:  Positions of positive integers in S'
A226137:  Positions of integers in S'

			S'= (1/1, 2/1, -1/1, 3/1, -1/2, 0/1, 4/1, -1/3, 1/2, ...), with integers appearing in positions 1,2,3,4,6,7,...

Clark Kimberling, Table of n, a(n) for n = 1..48

Cf. A226080 (rabbit ordering of positive rationals).

g[1] := {1}; z = 20; g[n_] := g[n] = DeleteCases[Flatten[Transpose[{# + 1, -1/#}]]&[DeleteCases[g[n - 1], 0]], Apply[Alternatives, Flatten[Map[g, Range[n - 1]]]]]; Flatten[Map[g, Range[7]]]  (* ordered rationals *)
Map[g, Range[z]]; Table[Length[g[i]], {i, 1, z}] (* cf A003410 *)
f = Flatten[Map[g, Range[z]]];
Take[Denominator[f], 100] (* A226130 *)
Take[Numerator[f], 100]   (* A226131 *)
p1 = Flatten[Table[Position[f, n], {n, 1, z}]] (* A226136 *)
p2 = Flatten[Table[Position[f, -n], {n, 0, z}]];
Union[p1, p2]  (* A226137 *)  (* Peter J. C. Moses, May 26 2013 *)

A003411 Losing initial positions in game: two players alternate in removing >= 1 stones; last player wins; first player may not remove all stones; each move <= 3 times previous move.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 11, 15, 21, 29, 40, 55, 76, 105, 145, 200, 276, 381, 526, 726, 1002, 1383, 1909, 2635, 3637, 5020, 6929, 9564, 13201, 18221, 25150, 34714, 47915, 66136, 91286, 126000, 173915, 240051, 331337, 457337, 631252, 871303, 1202640, 1659977

Offset: 0

N. J. A. Sloane, R. K. Guy, Rodney W. Topor (rwt(AT)cit.gu.edu.au)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. K. Guy, Letter to N. J. A. Sloane, Apr 1975
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 1).

Presumably equals A048590(n-3) - 3, n>3.

A003411:=-(1+z+z**2+z**3+z**4)/(-1+z+z**4); # Conjectured by Simon Plouffe in his 1992 dissertation

Join[{1}, LinearRecurrence[{1, 0, 0, 1}, {2, 3, 4, 6}, 80]] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)

a(n) = a(n-1) + a(n-4), n >= 5.

G.f.: (1+x+x^2+x^3+x^4)/(1-x-x^4).

A171855 Triangle read by rows: T(n,k) is the number of binary words of length n that have no pair of adjacent 1's and have k subwords 0000 (n>=0; k=0 for n=0,1,2; 0<=k<=n-3 for n>=3).

Original entry on oeis.org

1, 2, 3, 5, 7, 1, 10, 2, 1, 15, 3, 2, 1, 22, 6, 3, 2, 1, 32, 11, 6, 3, 2, 1, 47, 18, 12, 6, 3, 2, 1, 69, 30, 20, 13, 6, 3, 2, 1, 101, 50, 34, 22, 14, 6, 3, 2, 1, 148, 81, 59, 38, 24, 15, 6, 3, 2, 1, 217, 130, 99, 68, 42, 26, 16, 6, 3, 2, 1, 318, 208, 163, 118, 77, 46, 28, 17, 6, 3, 2, 1, 466

Offset: 0

Emeric Deutsch, Feb 15 2010

Row n has n-2 entries.
Sum of entries in row n is the Fibonacci number A000045(n+2).
T(n,0)=A003410(n). Sum(k*T(n,k),k>=0)=A004798(n-3) for n>=4.

			T(5,1)=2 because we have 00001 and 10000; T(7,2)=3 because we have 0000010, 0100000, and 1000001.
Triangle starts:
1;
2;
3;
5;
7,1;
10,2,1;
15,3,2,1;
22,6,3,2,1;

Cf. A000045, A003410, A004798.

G := (1+z)*(1+z-t*z+z^2-t*z^2+z^3-t*z^3)/(1-t*z-z^2+t*z^3-z^3+t*z^4-z^4): Gser := simplify(series(G, z = 0, 18)): for n from 0 to 15 do P[n] := sort(coeff(Gser, z, n)) end do: 1; 2; 3; for n from 3 to 15 do seq(coeff(P[n], t, k), k = 0 .. n-3) end do; # yields sequence in triangular form

G.f.: G(t,z) = (1+z)(1+z-tz+z^2-tz^2+z^3-tz^3)/(1-tz-z^2+tz^3-z^3+tz^4-z^4).

cp's OEIS Frontend

A038718 Number of permutations P of {1,2,...,n} such that P(1)=1 and |P^-1(i+1)-P^-1(i)| equals 1 or 2 for i=1,2,...,n-1.

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A097333 a(n) = Sum_{k=0..n} C(n-k, floor(k/2)).

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A226130 Denominators of rational numbers as generated by the rules: 1 is in S, and if nonzero x is in S, then x+1 and -1/x are in S. (See Comments.)

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A058278 Expansion of (1 - x^2)/(1 - x - x^3).

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A226131 Numerators of rational numbers as generated by the rules: 1 is in S, and if nonzero x is in S, then x+1 and -1/x are in S. (See Comments.)

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A101399 a(0) = 1, a(1) = 2, a(2) = 5; for n >= 3, a(n) = a(n-1) + 2*a(n-2) + a(n-3).

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A226136 Positions of the positive integers in the ordering of rational numbers as generated by the rules: 1 is in S, and if nonzero x is in S, then x+1 and -1/x are in S. (See Comments.)

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