A074049 -id:A074049 - cp's OEIS frontend

A183082 Tree generated by the Beatty sequence of 4-sqrt(6).

1, 2, 3, 5, 4, 8, 7, 14, 6, 11, 12, 22, 10, 19, 21, 39, 9, 16, 17, 30, 18, 33, 34, 61, 15, 28, 29, 53, 32, 59, 60, 109, 13, 25, 24, 45, 26, 47, 46, 84, 27, 50, 51, 92, 52, 95, 94, 171, 23, 42, 43, 78, 44, 81, 82, 149, 49, 90, 91, 166, 93, 168, 169, 306, 20

Offset: 1

Clark Kimberling, Dec 23 2010

nonn

A permutation of the positive integers.

			The top five rows:
1
2
3 5
4 8 7 14
6 11 12 22 10 19 21 39

Cf. A074049, A183079.

a = {1, 2}; row = {a[[-1]]}; r = 4 - Sqrt[6]; s = r/(r - 1); Do[a = Join[a, row = Flatten[{Floor[#*{r, s}]} & /@ row]], {n, 5}]; a (* Ivan Neretin, Nov 09 2015 *)

Let L(n)=floor(r*n) and U(n)=floor(s*n), where r=4-sqrt(6) and s=r/(r-1).
The tree-array T(n,k) is then given by rows:
T(0,0) = 1; T(1,0) = 2; T(n,2j) = L(T(n-1),j); T(n,2j+1) = U(T(n-1),j);
for j=0,1,...,2^(n-1)-1, n>=2.

A183083 Tree generated by the Beatty sequence of -1+sqrt(8).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 10, 13, 12, 15, 14, 17, 16, 19, 20, 24, 18, 22, 23, 28, 21, 26, 27, 33, 25, 30, 31, 37, 29, 35, 34, 41, 36, 44, 43, 52, 32, 39, 40, 48, 42, 50, 51, 61, 38, 46, 47, 57, 49, 59, 60, 72, 45, 55, 54, 66, 56, 68, 67, 81, 53, 64

Offset: 1

Clark Kimberling, Dec 23 2010

A permutation of the positive integers.

			Top five rows:
1
2
3 4
5 6 7 8
9 11 10 13 12 15 14 17

Cf. A074049, A183079.

a = {1, 2}; row = {a[[-1]]}; r = Sqrt[8] - 1; s = r/(r - 1); Do[a = Join[a, row = Flatten[{Floor[#*{r, s}]} & /@ row]], {n, 5}]; a (* Ivan Neretin, Nov 09 2015 *)

Let L(n)=Floor(r*n) and U(n)=Floor(s*n), where r=-1+sqrt(8) and s=r/(r-1).
The tree-array T(n,k) is then given by rows:
T(0,0) = 1; T(1,0) = 2; T(n,2j) = L(T(n-1),j); T(n,2j+1) = U(T(n-1),j);
for j=0,1,...,2^(n-1)-1, n>=2.

A183084 Tree generated by the Beatty sequence of e-1.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 9, 8, 11, 12, 16, 10, 14, 15, 21, 13, 19, 18, 26, 20, 28, 27, 38, 17, 23, 24, 33, 25, 35, 36, 50, 22, 31, 32, 45, 30, 43, 44, 62, 34, 47, 48, 66, 46, 64, 65, 90, 29, 40, 39, 55, 41, 57, 56, 78, 42, 59, 60, 83, 61, 86, 85, 119

Offset: 1

Clark Kimberling, Dec 23 2010

A permutation of the positive integers.

			Top 5 rows:
1
2
3 4
5 7 6 9
8 11 12 16 10 14 15 21

Cf. A074049, A183079.

a = {1, 2}; row = {a[[-1]]}; r = E - 1; s = r/(r - 1); Do[a = Join[a, row = Flatten[{Floor[#*{r, s}]} & /@ row]], {n, 5}]; a (* Ivan Neretin, Nov 09 2015 *)

Let L(n) = floor(r*n) and U(n) = floor(s*n), where r=e-1 and s=r/(r-1).
The tree-array T(n,k) is then given by rows:
T(0,0) = 1; T(1,0) = 2; T(n,2j) = L(T(n-1),j); T(n,2j+1) = U(T(n-1),j);
for j=0,1,...,2^(n-1)-1, n>=2.

A183090 Tree generated by A005652, associated with numbers which are not the sum of two Fibonacci numbers.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 7, 11, 12, 9, 10, 16, 15, 14, 13, 21, 23, 22, 25, 17, 18, 19, 20, 30, 33, 29, 31, 27, 28, 24, 26, 42, 41, 45, 46, 43, 44, 50, 49, 32, 34, 35, 36, 37, 38, 40, 39, 58, 60, 64, 67, 56, 59, 61, 62, 53, 54, 55, 57, 48, 47, 51, 52

Offset: 1

Clark Kimberling, Dec 24 2010

A permutation of the positive integers. See the comment at A183079.

			Top 5 rows:
  1;
  2;
  3,             4;
  6,      5,     8,      7;
  11, 12, 9, 10, 16, 15, 14, 13;
From row 3 to row 4: 3->(6,5) and 4->(8,7). For all such pairs, the 1st component is in L and the 2nd, in U.

Cf. A005652, A005653, A183079, A074049.

Let L(n)=A005652(n) and U(n)=A005653(n), these being complementary sequences, each comprising a maximal set no two of whose elements is a Fibonacci number.
The tree-array T(n,k) is then given by rows:
  T(0,0)=1; T(1,0)=2;
  T(n,2*j)=L(T(n-1,j));
  T(n,2*j+1)=U(T(n-1,j));
for j=0,1,...,2^(n-1)-1, n>=2.

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A183082 Tree generated by the Beatty sequence of 4-sqrt(6).

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A183083 Tree generated by the Beatty sequence of -1+sqrt(8).

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A183084 Tree generated by the Beatty sequence of e-1.

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A183090 Tree generated by A005652, associated with numbers which are not the sum of two Fibonacci numbers.

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