A242365 -id:A242365 - cp's OEIS frontend

A243736 Sum of the numbers in row n of the array at A242365.

1, 2, 4, 11, 27, 70, 185, 499, 1356, 3695, 10075, 27482, 74997, 204751, 559172, 1527379, 4172439, 11398634, 31140481, 85075307, 232426476, 634995031, 1734829379, 4739627674, 12948881373, 35376966263, 96651610708, 264057013451, 721417015071, 1970947675114

Offset: 1

Clark Kimberling, Jun 11 2014

Cf. A242364, A242365, A243735, A242448, A000045.

z = 12; g[1] = {1}; f1[x_] := 2 x; f2[x_] := 1 - x; f3[x_] := 2 - x; h[1] = g[1]; b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]], f3[g[n - 1]]]]; h[n_] := h[n] = Union[h[n - 1], g[n - 1]]; g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]; u = Table[g[n], {n, 1, 9}]
u1 = Flatten[u]  (* A242364 *)
v = Table[Reverse[Drop[g[n], Fibonacci[n - 1]]], {n, 1, z}]
v1 = Flatten[v]  (* A242365 *)
w1 = Table[Apply[Plus, g[n]], {n, 1, 20}]   (* A243735 *)
w2 = Table[Apply[Plus, v[[n]]], {n, 1, 10}] (* A243736 *)

First 7 rows of the array at A242365:
1
2
4
8 ... 3
16 .. 6 ... 5
32 .. 12 .. 10 .. 9 ... 7
64 .. 24 .. 20 .. 18 .. 17 .. 15 .. 14 .. 13;
the row sums are 1, 2, 4, 11, 27, 70 185,...

A226080 Denominators in the Fibonacci (or rabbit) ordering of the positive rational numbers.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 1, 4, 3, 2, 3, 1, 5, 4, 3, 4, 2, 5, 3, 1, 6, 5, 4, 5, 3, 7, 4, 2, 7, 5, 3, 5, 1, 7, 6, 5, 6, 4, 9, 5, 3, 10, 7, 4, 7, 2, 9, 7, 5, 7, 3, 8, 5, 1, 8, 7, 6, 7, 5, 11, 6, 4, 13, 9, 5, 9, 3, 13, 10, 7, 10, 4, 11, 7, 2, 11, 9, 7, 9, 5, 12, 7

Offset: 1

Clark Kimberling, May 25 2013

Let S be the set of numbers defined by these rules: 1 is in S, and if x is in S, then x+1 and 1/x are in S. Then S is the set of positive rational numbers, which arise in generations as follows: g(1) = (1/1), g(2) = (1+1) = (2), g(3) = (2+1, 1/2) = (3/1, 1/2), g(4) = (4/1, 1/3, 3/2), ... . Once g(n-1) = (g(1), ..., g(z)) is defined, g(n) is formed from the vector (g(1) + 1, 1/g(1), g(2) + 1, 1/g(2), ..., g(z) + 1, 1/g(z)) by deleting all elements that are in a previous generation. A226080 is the sequence of denominators formed by concatenating the generations g(1), g(2), g(3), ... . It is easy to prove the following:
(1) Every positive rational is in S.
(2) The number of terms in g(n) is the n-th Fibonacci number, F(n) = A000045(n).
(3) For n > 2, g(n) consists of F(n-2) numbers < 1 and F(n-1) numbers > 1, hence the name "rabbit ordering" since the n-th generation has F(n-2) reproducing pairs and F(n-1) non-reproducing pairs, as in the classical rabbit-reproduction introduction to Fibonacci numbers.
(4) The positions of integers in S are the Fibonacci numbers.
(5) The positions of 1/2, 3/2, 5/2, ..., are Lucas numbers (A000032).
(6) Continuing from (4) and (5), suppose that n > 0 and 0 < r < n, where gcd(n,r) = 1.  The positions in A226080 of the numbers congruent to r mod n comprise a row of the Wythoff array, W = A035513. The correspondence is sampled here:
   row 1 of W: positions of n+1 for n>=0,
   row 2 of W: positions of n+1/2,
   row 3 of W: positions of n+1/3,
   row 4 of W: positions of n+1/4,
   row 5 of W: positions of n+2/3,
   row 6 of W: positions of n+1/5,
   row 7 of W: positions of n+3/4.
(7) If the numbers <=1 in S are replaced by 1 and those >1 by 0, the resulting sequence is the infinite Fibonacci word A003849 (except for the 0-offset first term).
(8) The numbers <=1 in S occupy positions -1 + A001950, where A001950 is the upper Wythoff sequence; those > 1 occupy positions given by -1 + A000201, where A000201 is the lower Wythoff sequence.
(9) The rules (1 is in S, and if x is in S, then 1/x and 1/(x+1) are in S) also generate all the positive rationals.
A variant which extends this idea to an ordering of all rationals is described in A226130. - M. F. Hasler, Jun 03 2013
The updown and downup zigzag limits are (-1 + sqrt(5))/2 and (1 + sqrt(5))/2; see A020651. - Clark Kimberling, Nov 10 2013
From Clark Kimberling, Jun 19 2014: (Start)
Following is a guide to related trees and sequences; for example, the tree A226080 is represented by (1, x+1, 1/x), meaning that 1 is in S, and if x is in S, then x+1 and 1/x are in S (except for x = 0).
All the positive integers:
  A243571, A243572, A232559 (1, x+1, 2x)
  A232561, A242365, A243572 (1, x+1, 3x)
  A243573 (1, x+1, 4x)
All the integers:
  A243610 (1, 2x, 1-x)
  A232723, A242364
All the positive rationals:
  A226080, A226081, A242359, A242360 (1, x+1, 1/x)
  A243848, A243849, A243850 (1, x+1, 2/x)
  A243851, A243852, A243853 (1, x+1, 3/x)
  A243854, A243855, A243856 (1, x+1, 4/x)
  A243574, A242308 (1, 1/x, 1/(x+1))
  A241837, A243575 ({1,2,3}, x+4, 12/x)
  A242361, A242363 (1, 1 + 1/x, 1/x)
  A243613, A243614 (0, x+1, x/(x+1))
All the rationals:
  A243611, A243612 (0, x+1, -1/(x+1))
  A226130, A226131 (1, x+1, -1/x)
  A243712, A243713 ({1,2,3}, x+1, 1/(x+1))
  A243730, A243731 ({1,2,3,4}, x+1, 1/(x+1))
  A243732, A243733 ({1,2,3,4,5}, x+1, 1/(x+1))
  A243714, A243715
  A243925, A243926, A243927 (1, x+1, -2/x)
  A243928, A243929, A243930 (1, x+1, -3/x)
All the Gaussian integers:
  A243924 (1, x+1, i*x)
All the Gaussian rational numbers:
  A233694, A233695, A233696 (1, x+1, i*x, 1/x).
(End)

			The denominators are read from the rationals listed in "rabbit order":
1/1, 2/1, 3/1, 1/2, 4/1, 1/3, 3/2, 5/1, 1/4, 4/3, 5/2, 2/3, 6/1, ...

Clark Kimberling, Table of n, a(n) for n = 1..6000
Clark Kimberling, The infinite Fibonacci tree and other trees generated by rules, Proceedings of the 16th International Conference on Fibonacci Numbers and Their Applications, Fibonacci Quarterly 52 (2014), no. 5, pp. 136-149.
Index entries for fraction trees

Cf. A000045, A035513, A226081 (numerators), A226130, A226247, A020651.

z = 10; g[1] = {1}; g[2] = {2}; g[3] = {3, 1/2};
j[3] = Join[g[1], g[2], g[3]]; j[n_] := Join[j[n - 1], g[n]];
d[s_List, t_List] := Part[s, Sort[Flatten[Map[Position[s, #] &, Complement[s, t]]]]]; j[3] = Join[g[1], g[2], g[3]]; n = 3; While[n <= z, n++; g[n] = d[Riffle[g[n - 1] + 1, 1/g[n - 1]], g[n - 2]]];
Table[g[n], {n, 1, z}]; j[z] (* rabbit-ordered rationals *)
Denominator[j[z]]  (* A226080 *)
Numerator[j[z]]    (* A226081 *)
Flatten[NestList[(# /. x_ /; x > 1 -> Sequence[x, 1/x - 1]) + 1 &, {1}, 9]] (* rabbit-ordered rationals, Danny Marmer, Dec 07 2014 *)

A226080_vec(N=100)={my(T=[1],S=T,A=T); while(N>#A=concat(A, apply(denominator, T=select(t->!setsearch(S,t), concat(apply(t->[t+1,1/t],T))))), S=setunion(S,Set(T)));A} \\ M. F. Hasler, Nov 30 2018

(A226080(n)=denominator(RabbitOrderedRational(n))); ROR=List(1); RabbitOrderedRational(n)={if(n>#ROR, local(S=Set(ROR), i=#ROR*2\/(sqrt(5)+1), a(t)=setsearch(S,t)||S=setunion(S,[listput(ROR,t)])); until( type(ROR[i+=1])=="t_INT" && n<=#ROR, a(ROR[i]+1); a(1/ROR[i])));ROR[n]} \\ M. F. Hasler, Nov 30 2018

A242364 Irregular triangular array of all the integers ordered as in Comments.

Original entry on oeis.org

1, 0, 2, -1, 4, -3, -2, 3, 8, -7, -6, -4, 5, 6, 16, -15, -14, -12, -8, -5, 7, 9, 10, 12, 32, -31, -30, -28, -24, -16, -11, -10, -9, 13, 14, 15, 17, 18, 20, 24, 64, -63, -62, -60, -56, -48, -32, -23, -22, -20, -19, -18, -17, -13, 11, 25, 26, 28, 29, 30, 31

Offset: 1

Clark Kimberling, Jun 11 2014

Let f1(x) = 2x, f2(x) = 1-x, f3(x) = 2-x, g(1) = (1), and g(n) = union(f1(g(n-1)), f2(g(n-1)),f3(g(n-1))) for n >1. Let T be the array whose n-th row consists of the numbers in g(n) arranged in increasing order. It is easy to prove that every integer occurs exactly once in T. Conjectures: (1) |g(n)| = 2*F(n-1) for n >=2, where F = A000045 (the Fibonacci numbers), and exactly half of the numbers in g(n) are positive; (2) the number of even numbers in g(n) is 2*F(n-2) and the number of odd numbers is 2*F(n-3).

			First 6 rows of the array:
1
0 .... 2
-1 ... 4
-3 ... -2 .... 3 .... 8
-7 ... -6 ... -4 .... 5 .... 6 .... 16
-15 .. -14 .. -12 .. -8 .... -5 ... 7 ... 9 ... 10 ... 12 ... 32

Clark Kimberling, Table of n, a(n) for n = 1..3000

Cf. A242365, A243735, A243736, A242448, A000045.

z = 12; g[1] = {1}; f1[x_] := 2 x; f2[x_] := 1 - x; f3[x_] := 2 - x; h[1] = g[1]; b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]], f3[g[n - 1]]]]; h[n_] := h[n] = Union[h[n - 1], g[n - 1]]; g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]; u = Table[g[n], {n, 1, 9}]
u1 = Flatten[u]  (* A242364 *)
v = Table[Reverse[Drop[g[n], Fibonacci[n - 1]]], {n, 1, z}]
v1 = Flatten[v]  (* A242365 *)
w1 = Table[Apply[Plus, g[n]], {n, 1, 20}]   (* A243735 *)
w2 = Table[Apply[Plus, v[[n]]], {n, 1, 10}] (* A243736 *)

A242448 Number of distinct linear polynomials b+c*x in row n of array generated as in Comments.

Original entry on oeis.org

1, 3, 6, 12, 22, 38, 64, 106, 174, 284, 462, 750, 1216, 1970, 3190, 5164, 8358, 13526, 21888, 35418, 57310, 92732, 150046, 242782, 392832, 635618

Offset: 1

Clark Kimberling, Jun 11 2014

Let f1(x) = 2x, f2(x) = 1-x, f3(x) = 2-x, g(1) = (x), and g(n) = union(f1(g(n-1)), f2(g(n-1)),f3(g(n-1))) for n >1. Let T be the array whose n-th row consists of the polynomials b + c*x arranged by the relation << defined by b1 + c1*x << b2 + c2*x if c1 < c2, and b1 + c1*x << b2 + c2*x if c1 = c2 and b1 < b2. If x = 1, the array is as at A242364.

Apparently a(n) = A168193(n-1) for 3 <= n <= 26. - Georg Fischer, Oct 23 2018

			First 3 rows of the array of linear polynomials:
x .......................................... (1 polynomial)
1-x ... 2-x ... 2x ......................... (3 polynomials)
1-2x .. 2-2x .. 4-2x .. -1+x .. 1+x .. 4x .. (6 polynomials)

Cf. A168193, A242364, A242365.

z = 20; g[1] = {x}; f1[x_] := 2 x; f2[x_] := 1 - x; f3[x_] := 2 - x;
h[1] = g[1]; b[n_] := b[n] = Union[Expand[f1[g[n - 1]]], Expand[f2[g[n -
1]]], Expand[f3[g[n - 1]]]]; h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]];  u = Table[Length[g[n]], {n, 1, z}]  (* A242448 *)

Conjecture: a(n) = 2*a(n-1) - a(n-3) for n>= 6.

A243735 Sum of the numbers in row n of the array at A242364.

Original entry on oeis.org

1, 2, 3, 6, 10, 16, 26, 46, 88, 174, 342, 660, 1258, 2398, 4616, 8998, 17710, 35028, 69378, 137430, 272344, 540334, 1073798, 2137396, 4259770, 8496366, 16954536, 33843606, 67575358, 134965204

Offset: 1

Clark Kimberling, Jun 11 2014

			First 6 rows of the array at A242364:
1
0 .... 2
-1 ... 4
-3 ... -2 .... 3 .... 8
-7 ... -6 ... -4 .... 5 .... 6 .... 16
-15 .. -14 .. -12 .. -8 .... -5 ... 7 ... 9 ... 10 ... 12 ... 32;
the row sums are 1, 2, 3, 6, 10,16,...

Cf. A242364, A242365, A243736, A242448, A000045.

z = 12; g[1] = {1}; f1[x_] := 2 x; f2[x_] := 1 - x; f3[x_] := 2 - x; h[1] = g[1]; b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]], f3[g[n - 1]]]]; h[n_] := h[n] = Union[h[n - 1], g[n - 1]]; g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]; u = Table[g[n], {n, 1, 9}]
u1 = Flatten[u]  (* A242364 *)
v = Table[Reverse[Drop[g[n], Fibonacci[n - 1]]], {n, 1, z}]
v1 = Flatten[v]  (* A242365 *)
w1 = Table[Apply[Plus, g[n]], {n, 1, 20}]   (* A243735 *)
w2 = Table[Apply[Plus, v[[n]]], {n, 1, 10}] (* A243736 *)

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A243736 Sum of the numbers in row n of the array at A242365.

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A226080 Denominators in the Fibonacci (or rabbit) ordering of the positive rational numbers.

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A242364 Irregular triangular array of all the integers ordered as in Comments.

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A242448 Number of distinct linear polynomials b+c*x in row n of array generated as in Comments.

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A243735 Sum of the numbers in row n of the array at A242364.

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