A183171 -id:A183171 - cp's OEIS frontend

A183170 First of two trees generated by the Beatty sequence of sqrt(2).

1, 3, 4, 10, 5, 13, 14, 34, 7, 17, 18, 44, 19, 47, 48, 116, 9, 23, 24, 58, 25, 61, 62, 150, 26, 64, 66, 160, 67, 163, 164, 396, 12, 30, 32, 78, 33, 81, 82, 198, 35, 85, 86, 208, 87, 211, 212, 512, 36, 88, 90, 218, 93, 225, 226, 546, 94, 228

Offset: 1

Clark Kimberling, Dec 28 2010

This tree grows from (L(1),U(1))=(1,3). The other tree, A183171, grows from (L(2),U(2))=(2,6). Here, L is the Beatty sequence A001951 of r=sqrt(2); U is the Beatty sequence A001952 of s=r/(r-1). The two trees are complementary; that is, every positive integer is in exactly one tree. (L and U are complementary, too.) The sequence formed by taking the terms of this tree in increasing order is A183172.

			First levels of the tree:
.......................1
.......................3
..............4...................10
.........5..........13........14........34
.......7..17......18..44....19..47....48..116

Ivan Neretin, Table of n, a(n) for n = 1..8192

Cf. A183171, A183172, A001951, A001952, A178528, A074049.

a = {1, 3}; row = {a[[-1]]}; r = Sqrt[2]; s = r/(r - 1); Do[a = Join[a, row = Flatten[{Floor[#*{r, s}]} & /@ row]], {n, 5}]; a (* Ivan Neretin, May 25 2015 *)

See the formula at A178528, but use r=sqrt(2) instead of r=sqrt(3).

A183211 First of two trees generated by floor[(3n-1)/2].

Original entry on oeis.org

1, 3, 4, 9, 5, 12, 13, 27, 7, 15, 17, 36, 19, 39, 40, 81, 10, 21, 22, 45, 25, 51, 53, 108, 28, 57, 58, 117, 59, 120, 121, 243, 14, 30, 31, 63, 32, 66, 67, 135, 37, 75, 76, 153, 79, 159, 161, 324, 41, 84, 85, 171, 86, 174, 175, 351, 88, 177

Offset: 1

Clark Kimberling, Dec 30 2010

nonn

This tree grows from (L(1),U(1))=(1,3). The second tree, A183212, grows from (L(2),U(2))=(2,6). Here, L(n)=floor[(3n-1)/2] and U(n)=3n. The two trees are complementary in the sense that every positive integer is in exactly one tree. The sequence formed by taking the terms of this tree in increasing order is A183213. Leftmost branch of this tree: A183207. Rightmost: A000244. See A183170 and A183171 for the two trees generated by the Beatty sequence of sqrt(2).

			First four levels of the tree:
.......................1
.......................3
..............4..................9
............5...12............13....27

Ivan Neretin, Table of n, a(n) for n = 1..8192

Cf. A183170, A183212, A001651, A008585, A183209, A183213.

a = {1, 3}; row = {a[[-1]]}; Do[a = Join[a, row = Flatten[{Quotient[3 # - 1, 2], 3 #} & /@ row]], {n, 5}]; a (* Ivan Neretin, May 25 2015 *)

See the formula at A183209, but use L(n)=floor[(3n-1)/2] and U(n)=3n instead of L(n)=floor(3n/2) and U(n)=3n-1.

A183172 Ordering of the numbers in set S generated by these rules: 1 is in S, and if n is in S then floor(rn) and floor(sn) are in S, where r=sqrt(2) and s=r/(r-1).

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 10, 12, 13, 14, 16, 17, 18, 19, 22, 23, 24, 25, 26, 30, 31, 32, 33, 34, 35, 36, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 54, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 75, 76, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 93

Offset: 1

Clark Kimberling, Dec 28 2010

nonn

The sequence results from flattening and sorting the tree at A183170. Complement of A183173, obtained from the tree at A183171.

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

Cf. A183170, A183173.

nn=100; t={1}; r=Sqrt[2]; s=r/(r-1); t0=t; While[t=Select[Union[t, Floor[r*t], Floor[s*t]], #<=nn&]; t0 != t, t0=t]; t

See A183170.

A183173 Ordering of the numbers in set S generated by these rules: 2 is in S, and if n is in S then floor(rn) and floor(sn) are in S, where r=sqrt(2) and s=r/(r-1).

Original entry on oeis.org

2, 6, 8, 11, 15, 20, 21, 27, 28, 29, 37, 38, 39, 41, 51, 52, 53, 55, 57, 68, 71, 72, 73, 74, 77, 80, 92, 95, 96, 99, 100, 101, 103, 104, 108, 113, 126, 129, 130, 133, 134, 135, 139, 140, 141, 142, 145, 147, 152, 159, 174, 177, 178, 180, 182

Offset: 1

Clark Kimberling, Dec 29 2010

nonn

The sequence results from flattening and sorting the tree at A183171. Complement of A183172, obtained from the tree at A183170.

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

Cf. A183171, A183172.

nn=200; t={2}; r=Sqrt[2]; s=r/(r-1); t0=t; While[t=Select[Union[t, Floor[r*t], Floor[s*t]], # <= nn &]; t0 != t, t0=t]; t

(See A183171, A183172.)

Missing term 113 inserted by John W. Layman, Dec 29 2010

cp's OEIS Frontend

A183170 First of two trees generated by the Beatty sequence of sqrt(2).

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A183211 First of two trees generated by floor[(3n-1)/2].

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A183172 Ordering of the numbers in set S generated by these rules: 1 is in S, and if n is in S then floor(r*n) and floor(s*n) are in S, where r=sqrt(2) and s=r/(r-1).

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A183173 Ordering of the numbers in set S generated by these rules: 2 is in S, and if n is in S then floor(r*n) and floor(s*n) are in S, where r=sqrt(2) and s=r/(r-1).

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A183172 Ordering of the numbers in set S generated by these rules: 1 is in S, and if n is in S then floor(rn) and floor(sn) are in S, where r=sqrt(2) and s=r/(r-1).

A183173 Ordering of the numbers in set S generated by these rules: 2 is in S, and if n is in S then floor(rn) and floor(sn) are in S, where r=sqrt(2) and s=r/(r-1).