A226080 -id:A226080 - cp's OEIS frontend

A243924 Irregular triangular array of taxicab norms of Gaussian integers in array G generated as at Comments.

0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 3, 3, 4, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 6, 6, 6

Offset: 1

Clark Kimberling, Jun 17 2014

An array G of Gaussian integers is generated as follows: (row 1) = (0), and for n >=2, row n consists of the numbers x+1 and then i*x, where duplicates are deleted as they occur. Every Gaussian integer occurs exactly once in G. The taxicab norm of a Gaussian integer b+c*i is the taxicab distance (also known as Manhattan distance) from 0 to b+c*i, given by |b|+|c|. The norms of numbers in row n are given here in nondecreasing order. Conjecture: the number of numbers in row n is 4n-13 for n >= 5.

			First 6 rows of G:
0
1
2 .. i
3 .. 2i .. i+1 ... -1
4 .. 3i .. 1+2i .. -2 .. i+2 .. -1+i . -i
5 .. 4i .. 1+3i .. -3 .. 2+2i . -2+i . -2i . i+3 . -1+2i . -1-i . 1-i
The corresponding taxicab norms follow:
0
1
1 2
1 2 2 3
2 2 1 3 3 3 4
3 3 2 3 2 4 2 4 4 4 5
Each row is then arranged in nondecreasing order:
0
1
1 2
1 2 2 3
1 2 2 3 3 3 4
2 2 2 3 3 3 4 4 4 4 5

Clark Kimberling, Table of n, a(n) for n = 1..2000

Cf. A233694, A226080.

z = 10; g[1] = {0}; f1[x_] := x + 1; f2[x_] := I*x; h[1] = g[1];
b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[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]]]
Table[g[n], {n, 1, z}] (* the array G *)
v = Table[Abs[Re[g[n]]] + Abs[Im[g[n]]], {n, 1, z}]
w = Map[Sort, v] (* A243924, rows *)
w1 = Flatten[w]  (* A243924, sequence *)

A245218 Decimal expansion of sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A001951, else f(n,x) = 1/x.

Original entry on oeis.org

3, 4, 3, 6, 4, 8, 4, 8, 4, 3, 0, 9, 8, 1, 3, 5, 1, 7, 8, 4, 6, 1, 0, 5, 3, 9, 0, 3, 9, 2, 4, 7, 1, 3, 5, 6, 5, 0, 0, 9, 8, 8, 1, 6, 0, 6, 7, 3, 7, 8, 3, 0, 5, 4, 3, 6, 5, 8, 6, 6, 6, 6, 0, 5, 1, 7, 6, 2, 7, 1, 0, 7, 9, 0, 7, 6, 9, 8, 6, 2, 6, 0, 4, 6, 1, 6

Offset: 1

Clark Kimberling, Jul 13 2014

See Comments at A245215.

			c = 3.43648484309813517846105390392471356500...  The first 12 numbers f(n,1) comprise S(12) = {1, 2, 3, 1/3, 4/3, 7/3, 3/7, 10/7, 17/7, 24/7, 7/24, 31/24}; max(S(12)) = 24/7 = 3.42857...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A226080 (infinite Fibonacci tree), A245215, A245217, A245219, A245223.

tmpRec = $RecursionLimit; $RecursionLimit = Infinity; u[x_] := u[x] = x + 1; d[x_] := d[x] = 1/x; r = Sqrt[2]; w = Table[Floor[k*r], {k, 2000}]; s[1] = 1; s[n_] := s[n] = If[MemberQ[w, n - 1], u[s[n - 1]], d[s[n - 1]]]; $RecursionLimit = tmpRec;
m = Max[N[Table[s[n], {n, 1, 4000}], 300]]
t = RealDigits[m]  (* A245217 *)
(* Peter J. C. Moses, Jul 04 2014 *)

a(n)*inf{f(n,1)} = 1.

A245221 Decimal expansion of sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A022838, else f(n,x) = 1/x.

Original entry on oeis.org

2, 7, 2, 0, 7, 6, 6, 4, 5, 0, 7, 2, 9, 4, 7, 5, 2, 9, 7, 5, 4, 6, 9, 5, 1, 7, 3, 4, 8, 1, 7, 1, 5, 1, 3, 2, 4, 2, 5, 4, 7, 4, 9, 7, 9, 6, 1, 7, 1, 4, 6, 4, 1, 6, 7, 9, 0, 0, 0, 8, 2, 8, 3, 6, 6, 8, 7, 6, 6, 2, 4, 2, 1, 2, 1, 6, 7, 7, 7, 9, 0, 9, 7, 7, 8, 6

Offset: 1

Clark Kimberling, Jul 14 2014

See Comments at A245215.

			c = 0.367543491184951248721260972541092540...  The first 12 numbers f(n,1) comprise S(12) = {1, 2, 1/2, 3/2, 2/3, 5/3, 8/3, 3/8, 11/8, 8/11, 19/11, 11/19}; min(S(12)) = 3/8 = 0.375... and max(S(12)) = 8/3 = 2.666...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A226080 (infinite Fibonacci tree), A245215, A245217, A245220, A245222.

tmpRec = $RecursionLimit; $RecursionLimit = Infinity; u[x_] := u[x] = x + 1; d[x_] := d[x] = 1/x; r = Sqrt[3]; w = Table[Floor[k*r], {k, 2000}]; s[1] = 1; s[n_] := s[n] = If[MemberQ[w, n - 1], u[s[n - 1]], d[s[n - 1]]]; $RecursionLimit = tmpRec;
m = Max[N[Table[s[n], {n, 1, 4000}], 300]]
t = RealDigits[m]  (* A245221 *)
(* Peter J. C. Moses, Jul 04 2014 *)

a(n)*inf{f(n,1)} = 1.

A245224 Decimal expansion of sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A054385, else f(n,x) = 1/x.

Original entry on oeis.org

2, 7, 0, 7, 7, 7, 8, 7, 1, 6, 0, 0, 5, 0, 7, 8, 1, 2, 4, 3, 4, 0, 2, 0, 6, 6, 6, 5, 9, 6, 3, 1, 3, 1, 6, 2, 9, 9, 2, 3, 3, 1, 2, 4, 2, 4, 9, 1, 0, 4, 4, 5, 1, 7, 6, 6, 6, 9, 1, 3, 7, 9, 1, 8, 3, 4, 6, 4, 8, 3, 0, 8, 8, 4, 3, 2, 3, 4, 7, 0, 0, 2, 3, 5, 5, 3

Offset: 1

Clark Kimberling, Jul 14 2014

See Comments at A245215.

			c = 2.7077787160050781243402066659631316299233...  The first 16 numbers f(n,1) comprise S(16) = {1, 2, 1/2, 3/2, 5/2, 2/5, 7/5, 12/5, 5/12, 17/12, 12/17, 29/17}; min(S(16)) = 17/46 = 0.36956... and max(S(12)) = 46/17 = 2.7058...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A226080 (infinite Fibonacci tree), A245215, A245217, A245220, A245224.

tmpRec = $RecursionLimit; $RecursionLimit = Infinity; u[x_] := u[x] = x + 1; d[x_] := d[x] = 1/x; r = E/(E-1); w = Table[Floor[k*r], {k, 2000}]; s[1] = 1; s[n_] := s[n] = If[MemberQ[w, n - 1], u[s[n - 1]], d[s[n - 1]]]; $RecursionLimit = tmpRec;
m = Max[N[Table[s[n], {n, 1, 4000}], 300]]
RealDigits[m]  (* A245224 *)
(* Peter J. C. Moses, Jul 04 2014 *)

a(n)*inf{f(n,1)} = 1.

A226208 Zeckendorf distance between n and n+1.

Original entry on oeis.org

1, 1, 2, 3, 2, 4, 5, 2, 4, 6, 2, 7, 2, 4, 6, 2, 8, 2, 4, 9, 2, 4, 6, 2, 8, 2, 4, 10, 2, 4, 6, 2, 11, 2, 4, 6, 2, 8, 2, 4, 10, 2, 4, 6, 2, 12, 2, 4, 6, 2, 8, 2, 4, 13, 2, 4, 6, 2, 8, 2, 4, 10, 2, 4, 6, 2, 12, 2, 4, 6, 2, 8, 2, 4, 14, 2, 4, 6, 2, 8, 2, 4, 10

Offset: 1

Clark Kimberling, May 31 2013

Zeckendorf distance is defined at A226207.

			7 = 5 + 2 -> 3 + 1 -> 2, and 8 -> 5 -> 3 -> 2.  The total number of Zeckendorf downshifts (i.e., arrows) is 5, so that a(7) = D(7,8) = 5.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A226080.

zeck[n_Integer] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, z = {}},    While[k > 1, If[t >= Fibonacci[k], AppendTo[z, 1]; t = t - Fibonacci[k], AppendTo[z, 0]]; k--]; If[n > 0 && z[[1]] == 0, Rest[z], z]]; d[n1_, n2_] := Module[{z1 = zeck[n1], z2 = zeck[n2]}, Length[z1] + Length[z2] - 2 (NestWhile[# + 1 &, 1, z1[[#]] == z2[[#]] &, 1, Min[{Length[z1], Length[z2]}]] - 1)]; lst = Map[d[#, # + 1] &, Range[100]] (* Peter J. C. Moses, May 30 2013 *)

A226211 Zeckendorf distance between n and 2*n.

Original entry on oeis.org

1, 1, 1, 4, 3, 5, 6, 5, 7, 7, 8, 8, 7, 7, 9, 9, 8, 10, 10, 10, 9, 9, 9, 11, 11, 11, 11, 10, 12, 12, 12, 12, 12, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 12, 12, 14, 14, 14, 14, 14, 14, 14, 14, 13, 13, 13, 13, 13, 13, 13, 15, 15, 15, 15, 15, 15, 15, 15, 15

Offset: 1

Clark Kimberling, May 31 2013

Zeckendorf distance is defined at A226207.

			11 = 8 + 3 -> 5 + 2 -> 3 + 1 -> 2, and 22 = 21 + 1 -> 13 -> 8 -> 5 -> 3 -> 2. The total number of Zeckendorf downshifts (i.e., arrows) is 8, so that a(11) = D(11,22) = 8.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A226080, A226207, A226212.

zeck[n_Integer] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, z = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[z, 1]; t = t - Fibonacci[k], AppendTo[z, 0]]; k--]; If[n > 0 && z[[1]] == 0, Rest[z], z]]; d[n1_, n2_] := Module[{z1 = zeck[n1], z2 = zeck[n2]}, Length[z1] + Length[z2] - 2 (NestWhile[# + 1 &, 1, z1[[#]] == z2[[#]] &, 1, Min[{Length[z1], Length[z2]}]] - 1)];
lst = Map[d[#, 2#] &, Range[100]] (* Peter J. C. Moses, May 30 2013 *)

A226212 Zeckendorf distance between n and floor(n/2).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 31 2013

Zeckendorf distance is defined at A226207.

			11 = 8 + 3 -> 5 + 2 -> 3 + 1 -> 2, and 5 -> 3 -> 2.  The total number of Zeckendorf downshifts (i.e., arrows) is 5, so that a(11) = D(11,5) = 5.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A226080, A226207, A226211.

zeck[n_Integer] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, z = {}},    While[k > 1, If[t >= Fibonacci[k], AppendTo[z, 1]; t = t - Fibonacci[k], AppendTo[z, 0]]; k--]; If[n > 0 && z[[1]] == 0, Rest[z], z]]; d[n1_, n2_] := Module[{z1 = zeck[n1], z2 = zeck[n2]}, Length[z1] + Length[z2] - 2 (NestWhile[# + 1 &, 1, z1[[#]] == z2[[#]] &, 1,
Min[{Length[z1], Length[z2]}]] - 1)]; lst = Map[d[#, Floor[#/2]] &, Range[100]] (* Peter J. C. Moses, May 30 2013 *)

A226271 Index of 1/n in the Fibonacci (or rabbit) ordering of the positive rationals.

Original entry on oeis.org

1, 4, 6, 9, 14, 22, 35, 56, 90, 145, 234, 378, 611, 988, 1598, 2585, 4182, 6766, 10947, 17712, 28658, 46369, 75026, 121394, 196419, 317812, 514230, 832041, 1346270, 2178310, 3524579, 5702888, 9227466, 14930353, 24157818, 39088170, 63245987, 102334156, 165580142

Offset: 1

M. F. Hasler, Jun 01 2013

The Fibonacci ordering of the rationals (cf. A226080) is the sequence of rationals produced from the initial vector [1] by appending iteratively the new rationals obtained by applying the map t-> (t+1, 1/t) to the vector (cf. example).

Apart from initial terms, the same as A001611=(1, 2, 2, 3, 4, 6,...), A020706=(4,6,9,...), A048577=(3, 4, 6, ...), A000381=(2, 3, 4, ...).

			Starting from the vector [1] and applying the map t->(1+t,1/t), we get [2,1] (but ignore the number 1 which already occurred earlier), then [3,1/2], then [4,1/3,3/2,2] (where we ignore 2), etc. This yields the sequence (1,2,3,1/2,4,1/3,3/2,5,1/4,4/3,5/2,2/3,....) The unit fractions 1=1/1, 1/2, 1/3, ... occur at positions 1,4,6,9,...

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

LinearRecurrence[{2,0,-1},{1,4,6,9},40] (* Harvey P. Dale, Feb 04 2016 *)

PARI
```
A226271(n)=if(n>1,fibonacci(n+2))+1
```

{k=1;print1(s=1,",");U=Set(g=[1]);for(n=1,9,U=setunion(U,Set(g=select(f->!setsearch(U,f), concat(apply(t->[t+1,k/t],g))))); for(i=1,#g,numerator(g[i])==1&&print1(s+i","));s+=#g)} \\ for illustrative purpose

Vec(-x*(2*x^3+2*x^2-2*x-1)/((x-1)*(x^2+x-1)) + O(x^50)) \\ Colin Barker, May 11 2016

a(n) = 2*a(n-1)-a(n-3) for n>4. G.f.: -x*(2*x^3+2*x^2-2*x-1) / ((x-1)*(x^2+x-1)). - Colin Barker, Jun 03 2013
a(n) = 1+(2^(-1-n)*((1-sqrt(5))^n*(-3+sqrt(5))+(1+sqrt(5))^n*(3+sqrt(5))))/sqrt(5) for n>1. - Colin Barker, May 11 2016
E.g.f.: -2*(1 + x) + exp(x) + (3*sqrt(5)*sinh(sqrt(5)*x/2) + 5*cosh(sqrt(5)*x/2))*exp(x/2)/5. - Ilya Gutkovskiy, May 11 2016

A245216 Decimal expansion of sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A000201, else f(n,x) = 1/x.

Original entry on oeis.org

2, 7, 2, 9, 9, 6, 7, 7, 4, 1, 5, 9, 9, 8, 0, 2, 4, 8, 7, 8, 9, 1, 6, 4, 6, 7, 7, 4, 8, 7, 5, 9, 0, 7, 5, 2, 1, 1, 4, 3, 7, 8, 4, 1, 1, 3, 5, 3, 7, 0, 3, 4, 6, 2, 5, 9, 8, 6, 9, 5, 2, 7, 2, 4, 5, 2, 9, 0, 0, 6, 8, 8, 6, 4, 9, 3, 2, 6, 4, 2, 8, 6, 8, 0, 0, 6

Offset: 1

Clark Kimberling, Jul 13 2014

Equivalently, f(n,x) = 1/(f(n-1,x) if n is in A001950 (upper Wythoff sequence, given by w(n) = floor[tau*n], where tau = (1 + sqrt(5))/2, the golden ratio) and f(n,x) = f(n-1) + 1 otherwise. Let c = sup{f(n,1)}. The continued fraction of c is [2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, ...], which appears to be identical to the Hofstadter eta-sequence at A006340. See Comments at A245215.

			c = 2.7299677415998024878916467748759075211...  The first 12 numbers f(n,1) comprise S(12) = {1, 2, 1/2, 3/2, 5/2, 2/5, 7/5, 5/7, 12/7, 19/7, 7/19, 26/19}; max(S(12)) = 19/7 = 2.71429...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A226080 (infinite Fibonacci tree), A006340, A245215, A245217, A245220, A245223.

tmpRec = $RecursionLimit; $RecursionLimit = Infinity; u[x_] := u[x] = x + 1; d[x_] := d[x] = 1/x; r = GoldenRatio; w = Table[Floor[k*r], {k, 2000}]; s[1] = 1; s[n_] := s[n] = If[MemberQ[w, n - 1], u[s[n - 1]], d[s[n - 1]]]; $RecursionLimit = tmpRec;
m = Max[N[Table[s[n], {n, 1, 4000}], 300]]
RealDigits[m]  (* A245216 *)
(* Peter J. C. Moses, Jul 04 2014 *)

inf{f(n,1)}*(2 + a(n)) = 1.

A245225 Continued fraction expansion of the constant c in A245224; c = sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A054385, else f(n,x) = 1/x.

Original entry on oeis.org

2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2

Offset: 0

Clark Kimberling, Jul 14 2014

See Comments at A245215.

			c = 2.70777871600507812434020666596313162... ; The first 16 numbers f(n,1) comprise S(16) = {1, 2, 1/2, 3/2, 5/2, 2/5, 7/5, 12/5, 5/12, 17/12, 12/17, 29/17}; max(S(16)) = 46/17, with continued fraction [2, 1, 2, 2, 2].

Cf. A226080 (infinite Fibonacci tree), A245217, A245219, A245222, A245224 (decimal expansion).

tmpRec = $RecursionLimit; $RecursionLimit = Infinity; u[x_] := u[x] = x + 1; d[x_] := d[x] = 1/x; r = E/(E-1); w = Table[Floor[k*r], {k, 2000}]; s[1] = 1; s[n_] := s[n] = If[MemberQ[w, n - 1], u[s[n - 1]], d[s[n - 1]]]; max = Max[N[Table[s[n], {n, 1, 3000}], 200]] (* A245224 *)
ContinuedFraction[max, 120] (* A245225 *)

Offset changed by Andrew Howroyd, Jul 07 2024

cp's OEIS Frontend

A243924 Irregular triangular array of taxicab norms of Gaussian integers in array G generated as at Comments.

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A245218 Decimal expansion of sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A001951, else f(n,x) = 1/x.

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A245221 Decimal expansion of sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A022838, else f(n,x) = 1/x.

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A245224 Decimal expansion of sup{f(n,1)}, where f(1,x) = x + 1 and thereafter f(n,x) = x + 1 if n is in A054385, else f(n,x) = 1/x.

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A226208 Zeckendorf distance between n and n+1.

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A226211 Zeckendorf distance between n and 2*n.

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A226212 Zeckendorf distance between n and floor(n/2).

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A226271 Index of 1/n in the Fibonacci (or rabbit) ordering of the positive rationals.

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