A274142 -id:A274142 - cp's OEIS frontend

A274159 Number of integers in n-th generation of tree T(3^(-1/3)) defined in Comments.

1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 6, 7, 9, 11, 12, 16, 18, 23, 28, 33, 41, 49, 61, 72, 89, 107, 130, 159, 191, 234, 283, 345, 418, 507, 617, 747, 910, 1103, 1340, 1629, 1976, 2402, 2914, 3542, 4300, 5223, 6344, 7701, 9359, 11361, 13801, 16761, 20353, 24725, 30021, 36468, 44285, 53788, 65328

Offset: 0

Clark Kimberling, Jun 12 2016

nonn

Let T* be the infinite tree with root 0 generated by these rules: if p is in T*, then p+1 is in T* and x*p is in T*. Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3,2x,x+1,x^2}, etc. Let T(r) be the tree obtained by substituting r for x.

See A274142 for a guide to related sequences.

			If r = 3^(-1/3), then g(3) = {3,2r,r+1, r^2}, in which the number of integers is a(3) = 1.

Kenny Lau, Table of n, a(n) for n = 0..11841

Cf. A274142.

A274159 := proc(r)
    local gs, n, gs2, el, a ;
    gs := [1] ;
    for n from 2 do
        gs2 := [] ;
        for el in gs do
            gs2 := [op(gs2), el+1, r*el] ;
        end do:
        gs := gs2 ;
        a := 0 ;
        for el in gs do
            if type(el, 'integer') then
                 a := a+1 :
            end if;
        end do:
        print(n, a) ;
    end do:
end proc:
A274159(1/root[3](3)) ; # R. J. Mathar, Jun 20 2016

z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
u = Table[t[[k]] /. x -> 3^(-1/3), {k, 1, z}]; Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]

a(15)-a(18) from R. J. Mathar, Jun 20 2016

More terms from Kenny Lau, Jul 04 2016

A274165 Number of real integers in n-th generation of tree T(i/3) defined in Comments.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 17, 21, 26, 32, 39, 47, 57, 67, 79, 93, 110, 131, 157, 189, 228, 276, 332, 399, 478, 571, 681, 812, 969, 1158, 1387, 1662, 1994, 2393, 2871, 3442, 4123, 4935, 5904, 7063, 8449, 10111

Offset: 0

Clark Kimberling, Jun 12 2016

nonn

Let T* be the infinite tree with root 0 generated by these rules:  if p is in T*, then p+1 is in T* and x*p is in T*.  Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3,2x,x+1,x^2}, etc.  Let T(r) be the tree obtained by substituting r for x.
See A274142 for a guide to related sequences.
a(n) = A017885(n+7) for 2 <= n < 85, but a(85) = 1314173 differs from A017885(92) = 1314172. - Georg Fischer, Oct 30 2018

			If r = i/3, then g(3) = {3,2r,r+1, r^2}, in which the number of real integers is a(3) = 1.

Kenny Lau, Table of n, a(n) for n = 0..4727

Cf. A274142.

z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
u = Table[t[[k]] /. x -> I/3, {k, 1, z}]; Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]

More terms from Kenny Lau, Jun 30 2017

A274185 Irregular triangular array having n-th row g(n) defined in Comments.

Original entry on oeis.org

0, 1, 2, 0, 3, 1, 4, 2, 1, 0, 5, 3, 2, 1, 0, 6, 4, 3, 2, 1, 2, 1, 0, 7, 5, 4, 3, 2, 3, 2, 1, 1, 0, 0, 8, 6, 5, 4, 3, 4, 3, 2, 2, 1, 1, 3, 2, 1, 1, 0, 0, 9, 7, 6, 5, 4, 5, 4, 3, 3, 2, 2, 4, 3, 2, 2, 1, 1, 2, 1, 1, 0, 0, 1, 0, 0, 10, 8, 7, 6, 5, 6, 5, 4, 4, 3

Offset: 0

Clark Kimberling, Jun 13 2016

Let g(0) = (0) and for n > 0, define g(n) inductively to be the concatenation of g(n-1) and the numbers (k-1)/2 as k ranges through the odd numbers k in g(n-1). Every nonnegative integer appears infinitely many times. For the limiting ratio of lengths of consecutive rows, see A274192.

			First seven rows:
0
1
2   0
3   1
4   2   1   0
5   3   2   1   0
6   4   3   2   1   2   1   0

Cf. A274142 (row lengths), A274192, A274183.

g[0] = {0}; z = 14; g[n_] := g[n] = Join[g[n - 1] + 1, (1/2) (Select[g[n - 1], IntegerQ[(# - 1)/2] &] - 1)]; Flatten[Table[g[n], {n, 0, z}]]

A274153 Number of integers in n-th generation of tree T(5/2) defined in Comments.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 8, 12, 19, 27, 41, 60, 90, 136, 204, 306, 461, 692, 1036, 1557, 2330, 3498, 5247, 7872, 11803, 17715, 26558, 39851, 59781, 89651, 134482, 201736, 302605, 453892, 680800, 1021307, 1531944, 2297678, 3446694, 5170101, 7754719, 11632249, 17448162, 26172247, 39258668, 58886868, 88331299, 132497660

Offset: 0

Clark Kimberling, Jun 12 2016

Let T* be the infinite tree with root 0 generated by these rules: if p is in T*, then p+1 is in T* and x*p is in T*. Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3,2x,x+1,x^2}, etc. Let T(r) be the tree obtained by substituting r for x.

See A274142 for a guide to related sequences.

			For r = 5/2, we have g(3) = {3,2r,r+1, r^2}, in which the number of integers is a(3) = 2.

Cf. A274142.

z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
u = Table[t[[k]] /. x -> 5/2, {k, 1, z}]; Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]

More terms from Kenny Lau, Jul 02 2016

A274162 Number of real integers in n-th generation of tree T(3i) defined in Comments.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 234, 379, 615, 997, 1617, 2622, 4252, 6895, 11181

Offset: 0

Clark Kimberling, Jun 12 2016

Let T* be the infinite tree with root 0 generated by these rules: if p is in T*, then p+1 is in T* and x*p is in T*. Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3,2x,x+1,x^2}, etc. Let T(r) be the tree obtained by substituting r for x.

See A274142 for a guide to related sequences.

			If r = 3i, then g(3) = {3,2r,r+1, r^2}, in which the number of real integers is a(3) = 2.

Cf. A274142.

z = 22; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
u = Table[t[[k]] /. x -> 3 I, {k, 1, z}]; Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]

a(19)-a(21) from Ryan Hitchman, Sep 13 2017

A274163 Number of real integers in n-th generation of tree T(4i) defined in Comments.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6766, 10948, 17716, 28667, 46388, 75063

Offset: 0

Clark Kimberling, Jun 12 2016

Let T* be the infinite tree with root 0 generated by these rules:  if p is in T*, then p+1 is in T* and x*p is in T*.  Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3,2x,x+1,x^2}, etc.  Let T(r) be the tree obtained by substituting r for x.
For each integer k > 0, let s(k,n) be the number of integers in the n-th generation of T(k*i).  Conjecture:  there is a limiting sequence S(n) as k increases, and S(n) = F(n) for n >= 1, where F = A000045 (Fibonacci numbers).
From Charlie Neder, Jul 11 2018: (Start)
Assume for the moment that a complex number cannot be transformed back into an integer. If this is the case, then the real integers in g(n) are the real integers in g(n-1) plus 1 and the imaginary integers in g(n-1) times k*i, which are themselves k*i times the real integers in g(n-2), and so S(n) = S(n-1) + S(n-2) and S(n) = F(n).
However, the above assumption is false, but the earliest time such a transformation can take place is at g(k^2+5), following this path: 0 -> 1 -> k*i -> 1+k*i -> -k^2+k*i -> -(k^2-1)+k*i -> ... -> k*i -> -k^2.
Therefore s(k,n) matches the Fibonacci sequence for n < k^2+5 and S(n) = F(n). (End)
a(n) = A000045(n) only for 0 < n < 21. - Robert G. Wilson v, Jul 23 2018

			If r = 4i, then g(3) = {3,2r,r+1, r^2}, in which the number of real integers is a(3) = 2.

See A274142 for a guide to related sequences.

z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
u = Table[t[[k]] /. x -> 4 I, {k, 1, z}]; Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]

a(21)-a(25) from Robert G. Wilson v, Jul 23 2018

A274166 Number of real integers in n-th generation of tree T(i+1) defined in Comments.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 8, 15, 24, 44, 84, 146, 254, 443, 761, 1317, 2262

Offset: 0

Clark Kimberling, Jun 13 2016

Let T* be the infinite tree with root 0 generated by these rules: if p is in T*, then p+1 is in T* and x*p is in T*. Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3,2x,x+1,x^2}, etc. Let T(r) be the tree obtained by substituting r for x.

See A274142 for a guide to related sequences.

			If r = i+1, then g(3) = {3,2r,r+1, r^2}, in which the number of real integers is a(3) = 1.

Cf. A274142.

z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
u = Table[t[[k]] /. x -> I + 1, {k, 1, z}]; Table[
Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]

A274167 Number of real integers in n-th generation of tree T(i-1) defined in Comments.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 5, 8, 14, 24, 39, 67, 112, 191, 327, 560, 956, 1635, 2796

Offset: 0

Clark Kimberling, Jun 13 2016

Let T* be the infinite tree with root 0 generated by these rules: if p is in T*, then p+1 is in T* and x*p is in T*. Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3,2x,x+1,x^2}, etc. Let T(r) be the tree obtained by substituting r for x.

See A274142 for a guide to related sequences.

			If r = i-1, then g(3) = {3,2r,r+1, r^2}, in which the number of real integers is a(3) = 1.

Cf. A274142.

z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
u = Table[t[[k]] /. x -> I - 1, {k, 1, z}]; Table[
Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]

A274168 Number of real integers in n-th generation of tree T(r) defined in Comments, where r^2 = -r - 1 (i.e., r = (-1 + sqrt(3))/2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 9, 13, 18, 25, 33, 43

Offset: 0

Clark Kimberling, Jun 13 2016

Let T* be the infinite tree with root 0 generated by these rules: if p is in T*, then p+1 is in T* and x*p is in T*. Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3,2x,x+1,x^2}, etc. Let T(r) be the tree obtained by substituting r for x.

See A274142 for a guide to related sequences.

			If r = (-1 + sqrt(3))/2, then g(3) = {3,2r,r+1,r^2}, in which the number of real integers is a(3) = 1.

Cf. A274142.

z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
u = Table[t[[k]] /. x -> (-1 + 3 I)/2, {k, 1, z}]; Table[
Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]

cp's OEIS Frontend

A274159 Number of integers in n-th generation of tree T(3^(-1/3)) defined in Comments.

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A274165 Number of real integers in n-th generation of tree T(i/3) defined in Comments.

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A274185 Irregular triangular array having n-th row g(n) defined in Comments.

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A274153 Number of integers in n-th generation of tree T(5/2) defined in Comments.

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A274162 Number of real integers in n-th generation of tree T(3i) defined in Comments.

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A274163 Number of real integers in n-th generation of tree T(4i) defined in Comments.

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A274166 Number of real integers in n-th generation of tree T(i+1) defined in Comments.

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A274167 Number of real integers in n-th generation of tree T(i-1) defined in Comments.

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