A141832 -id:A141832 - cp's OEIS frontend

A228853 Nodes of tree generated as follows: (1,2) is an edge, and if (x,y) is an edge, then (y,y+x) and (y,2y+x) are edges.

1, 2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 21, 26, 27, 29, 30, 31, 34, 41, 43, 44, 45, 46, 47, 49, 50, 55, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 79, 80, 81, 89, 97, 99, 100, 101, 104, 105, 106, 108, 109, 111, 112, 115, 116, 117, 119, 121, 123, 128

Offset: 1

Clark Kimberling, Sep 05 2013

As a tree, infinitely many branches are essentially linearly recurrent sequences.  The extreme cases, (1,2) -> (2,3) -> (3,5) -> ... and (1,2) -> (2,5) -> (5,12) -> ..., contribute A000045 (Fibonacci numbers) and A000129 (Pell numbers) to A228853.
Suppose that (u,v) and (v,w) are consecutive edges.  The continued fraction of w/v is obtained from the continued fraction of v/u by prefixing 1 if w = v + u, or 2 if w = 2v + u. Consequently, if each edge is labeled with 1 or 2 in the obvious way, then the continued fraction of w/v is the sequence of 1s and 2s, in reverse order, from the node 2 to the node w, with 2 attached at the end. (See Example, Part 2.)
Is A228853 essentially A141832?  (If so, the answer to the question in Comments at A141832 is that A141832 is infinite.)
Yes; the initial node (1,2) adds a single 2 to the end of the fraction, and subsequent edges prepend 1's and 2's. - Charlie Neder, Oct 21 2018

			Part 1: Taking the first generation of edges of the tree to be G(1) = {(1,2)}, the edge (1,2) grows G(2) = {(2,3), (2,5)}, which grows G(3) = {(3,5), (3,8), (5,7), (5,12)}, ... Expelling duplicate nodes and sorting leave {1, 2, 3, 5, 7, 8, ...}.
Part 2: The branch 2, 3, 8, 11, 19, 30, 49, 128, 305 has edge-labels 1, 2, 1, 1, 1, 1, 2, 2, so that 305/128 = [2, 2, 1, 1, 1, 1, 2, 1, 2].

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A141832, A228854, A228855, A228856, A000045, A000129.

f[x_, y_] := {{y, x + y}, {y, x + 2 y}}; x = 1; y = 2; t = {{x, y}}; u = Table[t = Flatten[Map[Apply[f, #] &, t], 1], {12}]; v = Flatten[u]; w = Flatten[Prepend[Table[v[[2 k]], {k, 1, Length[v]/2}], {x, y}]]; Sort[Union[w]]

A141822 Maximum term in the continued fraction of A141821(n)/n.

Original entry on oeis.org

2, 2, 3, 2, 5, 2, 2, 3, 3, 2, 2, 2, 3, 3, 3, 2, 2, 2, 4, 2, 3, 3, 3, 3, 2, 2, 4, 2, 2, 2, 3, 3, 2, 3, 3, 3, 4, 3, 3, 2, 4, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 5, 2, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 2, 4, 3, 3, 3, 3, 3, 4, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2

Offset: 2

T. D. Noe, Jul 08 2008

nonn

Consider the continued fraction [0;c1,c2,...,cm] of k/n, with k

Zaremba conjectured that a(n)<=5, a bound that is attained for n in A195901. It appears that n=150 may be the largest integer with a(n)=5, while n=6234 may be the largest integer with a(n)=4.

Robin Visser, Table of n, a(n) for n = 2..10000 (terms n = 2..2000 from T. D. Noe).

See A141821 for the least value of k for each n.
See A141832, A141833, A141823, and A195901 for the integers n>1 such that a(n) = 2, 3, 4, and 5, respectively.
Cf. A006839 (where cm is constrained to be 1).

Table[c=ContinuedFraction[Select[Range[n-1],GCD[ #,n]==1&]/n]; Min[Max/@c], {n, 150}]

vecmax(v)=my(mx=v[1]); for(i=2,#v,mx=max(mx,v[i])); mx
a(n)=vecmin([vecmax(contfrac(k/n))|k<-[1..n],gcd(k,n)==1]) \\ Charles R Greathouse IV, Jul 18 2014

Edited by Max Alekseyev, Sep 25 2011

A141833 Integers n>1 such that A141822(n)=3.

Original entry on oeis.org

4, 9, 10, 14, 15, 16, 22, 23, 24, 25, 32, 33, 35, 36, 37, 39, 40, 48, 51, 52, 53, 56, 57, 58, 59, 60, 61, 62, 66, 72, 77, 78, 82, 83, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 98, 102, 103, 107, 110, 113, 114, 118, 120, 122, 124, 125, 126, 127, 130, 132, 133, 134, 135, 136

Offset: 1

T. D. Noe, Jul 09 2008

nonn

Zaremba conjectured that A141823, A141832, A195901, and this sequence form a partition of the integers >1.Is this sequence finite or infinite?

T. D. Noe, Table of n, a(n) for n=1..5563

Cf. A141821.

A228856 Nodes of tree generated as follows: (1,2) is an edge, and if (x,y) is an edge, then (y,y+x), (y,2y+x), and (y,3y+x) are edges.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 39, 41, 43, 44, 45, 46, 47, 49, 50, 52, 53, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 83, 84

Offset: 1

Clark Kimberling, Sep 05 2013

			Taking the first generation of edges of the tree to be G(1) = {(1,2)}, the edge (1,2) grows G(2) = {(2,3), (2,5), (2,7)}, which grows G(3) = {(3,5), (3,8), (3,11), (5,7), (5,12), (5,17), (7,9), (7,16), (7,23)}, ... Expelling duplicate nodes and sorting leave {1,2,3,5,7,8,9,...}.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A141832, A228853, A228854, A228856.

f[x_, y_] := {{y, x + y}, {y, x + 2 y}}; x = 2; y = 3; t = {{x, y}};
u = Table[t = Flatten[Map[Apply[f, #] &, t], 1], {12}]; v = Flatten[u];
w = Flatten[Prepend[Table[v[[2 k]], {k, 1, Length[v]/2}], {x, y}]];
Sort[Union[w]]

A141823 Integers n>1 such that A141822(n)=4.

Original entry on oeis.org

20, 28, 38, 42, 90, 96, 156, 164, 216, 228, 252, 318, 336, 350, 384, 386, 442, 508, 558, 770, 876, 922, 978, 1014, 1155, 1170, 1410, 1450, 1692, 1870, 2052, 2370, 3618, 5052, 6234

Offset: 1

T. D. Noe, Jul 08 2008, Jul 09 2008

No other terms below 10^5.
Zaremba conjectured that A141832, A141833, A195901, and this sequence form a partition of the integers >1.
Is this sequence finite? If so, is it complete?

Cf. A141832, A141833, A141821.

Edited by Max Alekseyev, Sep 25 2011

A195901 Integers n>1 such that A141822(n)=5.

Original entry on oeis.org

6, 54, 150

Offset: 1

Max Alekseyev, Sep 25 2011

No other terms below 10^6.
Zaremba conjectured that A141823, A141832, A141833, and this sequence form a partition of the integers >1.
Is this sequence finite? If so, is it complete?

A. Kontorovich, From Apollonius to Zaremba: Local-global phenomena in thin orbits, Bull. Amer. Math. Soc., 50 (2013), 187-228. - From _N. J. A. Sloane_, Dec 25 2012

Cf. A141823, A141832, A141833.

A228854 Nodes of tree generated as follows: (1,3) is an edge, and if (x,y) is an edge, then (y,y+x) and (y,2y+x) are edges.

Original entry on oeis.org

1, 3, 4, 7, 10, 11, 15, 17, 18, 24, 25, 26, 27, 29, 37, 40, 41, 43, 44, 47, 56, 58, 61, 63, 64, 65, 67, 68, 69, 71, 76, 89, 91, 93, 97, 98, 99, 100, 101, 104, 105, 106, 108, 109, 111, 112, 115, 123, 137, 138, 140, 147, 149, 152, 153, 154, 155, 157, 159, 160

Offset: 1

Clark Kimberling, Sep 05 2013

			Taking the first generation of edges of the tree to be G(1) = {(1,3)}, the edge (1,3) grows G(2) = {(3,4), (3,7)}, which grows G(3) = {(4,7), (4,11), (7,10),(7,17)}, ... Expelling duplicate nodes and sorting leave {1,3,4,7,10,11,...}.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A141832, A228853, A228855, A228856.

f[x_, y_] := {{y, x + y}, {y, x + 2 y}}; x = 1; y = 3; t = {{x, y}};
u = Table[t = Flatten[Map[Apply[f, #] &, t], 1], {12}]; v = Flatten[u];
w = Flatten[Prepend[Table[v[[2 k]], {k, 1, Length[v]/2}], {x, y}]];
Sort[Union[w]]

A228855 Nodes of tree generated as follows: (2,3) is an edge, and if (x,y) is an edge, then (y,y+x) and (y,2y+x) are edges.

Original entry on oeis.org

2, 3, 5, 8, 11, 13, 18, 19, 21, 27, 29, 30, 31, 34, 41, 44, 46, 47, 49, 50, 55, 65, 67, 68, 71, 73, 75, 76, 79, 80, 81, 89, 100, 101, 106, 108, 109, 111, 112, 115, 116, 117, 119, 121, 123, 128, 129, 131, 144, 153, 157, 163, 165, 166, 171, 172, 173, 175, 176

Offset: 1

Clark Kimberling, Sep 05 2013

			Taking the first generation of edges of the tree to be G(1) = {(2,3)}, the edge (2,3) grows G(2) = {(3,5), (3,8)}, which grows G(3) = {(5,8), (5,13), (8,11),(8,19)}, ... Expelling duplicate nodes and sorting leave {2,3,5,8,11,13,...}.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A141832, A228853, A228854, A228856.

f[x_, y_] := {{y, x + y}, {y, x + 2 y}}; x = 2; y = 3; t = {{x, y}};
u = Table[t = Flatten[Map[Apply[f, #] &, t], 1], {12}]; v = Flatten[u];
w = Flatten[Prepend[Table[v[[2 k]], {k, 1, Length[v]/2}], {x, y}]];
Sort[Union[w]]

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A228853 Nodes of tree generated as follows: (1,2) is an edge, and if (x,y) is an edge, then (y,y+x) and (y,2y+x) are edges.

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A141822 Maximum term in the continued fraction of A141821(n)/n.

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A141833 Integers n>1 such that A141822(n)=3.

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A228856 Nodes of tree generated as follows: (1,2) is an edge, and if (x,y) is an edge, then (y,y+x), (y,2y+x), and (y,3y+x) are edges.

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A141823 Integers n>1 such that A141822(n)=4.

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A195901 Integers n>1 such that A141822(n)=5.

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A228854 Nodes of tree generated as follows: (1,3) is an edge, and if (x,y) is an edge, then (y,y+x) and (y,2y+x) are edges.

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A228855 Nodes of tree generated as follows: (2,3) is an edge, and if (x,y) is an edge, then (y,y+x) and (y,2y+x) are edges.

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