A191329 -id:A191329 - cp's OEIS frontend

A191330 Positions of 0 in A191329.

4, 8, 10, 14, 20, 24, 26, 30, 36, 40, 46, 50, 52, 56, 62, 66, 68, 72, 76, 78, 82, 88, 92, 94, 98, 104, 108, 114, 118, 120, 124, 130, 134, 136, 140, 144, 146, 150, 156, 160, 162, 166, 172, 176, 178, 182, 186, 188, 192, 198, 202, 204, 208, 214, 218, 224, 228, 230, 234, 240, 244, 246, 250, 254, 256, 260, 266, 270, 272, 276, 282, 286

Offset: 1

Clark Kimberling, May 31 2011

nonn

A191330=2*A005653.

			A191329=(1,2,1,0,1,2,1,0,1,0,1,2,1,0,1...),
so that a(1)=4, a(2)=8, a(3)=10,...

Cf. A191329, A005653.

Mathematica
```
(See A191329.)
```

A191331 Positions of 2 in A191329.

Original entry on oeis.org

2, 6, 12, 16, 18, 22, 28, 32, 34, 38, 42, 44, 48, 54, 58, 60, 64, 70, 74, 80, 84, 86, 90, 96, 100, 102, 106, 110, 112, 116, 122, 126, 128, 132, 138, 142, 148, 152, 154, 158, 164, 168, 170, 174, 180, 184, 190, 194, 196, 200, 206, 210, 212, 216, 220, 222, 226, 232, 236, 238, 242, 248, 252, 258, 262, 264, 268, 274, 278, 280, 284, 288

Offset: 1

Clark Kimberling, May 31 2011

nonn

A191331=2*A005652.

			A191329=(1,2,1,0,1,2,1,0,1,0,1,2,...), so that
a(1)=2, a(2)=6, a(3)=12,...

Cf. A191329, A191330, A191332, A191333.

Mathematica
```
(See A191329.)
```

A191336 (A022838 mod 2)+(A054406 mod 2).

Original entry on oeis.org

1, 1, 2, 1, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 1, 2, 1, 1, 2, 1, 1, 0, 1, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 1, 2, 1, 1, 2, 1, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 1, 2, 1, 1, 2, 1, 1, 0, 1, 1, 2, 2, 1, 0

Offset: 1

Clark Kimberling, Jun 01 2011

nonn

A022838: Beatty sequence for r=sqrt(3),
A054406: Beatty sequence for s=(3+sqrt(3))/2 (complement
of A022838), so that
A191336(n)=([nr] mod 2)+([ns] mod 2), where [ ]=floor.
A191336(n)=(number of odd numbers in {[nr],[ns]}).

Cf. A191329, A191337, A191338, A191339.

r = Sqrt[3]; s = r/(r - 1); h = 320;
u = Table[Floor[n*r], {n, 1, h}] (* A022838 *)
v = Table[Floor[n*s], {n, 1, h}] (* A054406 *)
w = Mod[u, 2] + Mod[v, 2] (* A191336 *)
Flatten[Position[w, 0]]   (* A191337 *)
Flatten[Position[w, 1]]   (* A191338 *)
Flatten[Position[w, 2]]   (* A191339 *)

a(n)=([nr] mod 2)+([ns] mod 2), where r=sqrt(3), s=r/(r-1), and [ ]=floor.

A191332 Decimal expansion of sum{(1/3)^A191330(k): k>=1}.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, May 31 2011

sum{(1/3)^A191330(k)}+sum{(1/3)^A191331(k)}= 1/8.

			0.01251523925632429136544313170551950...

Cf. A191329, A191330, A191331, A191332, A191333.

Mathematica
```
(See A191329.)
```

A191340 (A022839 mod 2)+(A108598 mod 2).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 0, 0, 1, 1, 2, 2, 2, 1, 0, 0, 0, 0, 1, 2, 2, 2, 2, 1, 0, 0, 0, 1, 1, 2, 2, 1, 1, 1, 0, 0, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 0, 0, 1, 1, 1, 2, 2, 1, 1, 0, 0, 0, 1, 2, 2, 2, 2, 1, 0, 0, 0, 0, 1, 2, 2, 2, 1, 1, 0, 0, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 1, 2, 2, 1, 1, 0, 0, 0, 1, 2, 2, 2, 2, 1, 0, 0, 0, 0, 1, 2, 2, 2, 1, 1, 0, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 1

Offset: 1

Clark Kimberling, Jun 01 2011

nonn

Let r=sqrt(5) and s=r/(r-1).  There numbers yield the following two complementary Beatty sequences:
A022839(n)=[nr], A108598(n)=[ns], where [ ]=floor.
A191340(n)=the number of odd numbers in {[nr], [ns]}.

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

Cf. A191329, A191336, A191380, A191381, A191382.

r = Sqrt[5]; s = r/(r - 1); h = 120;
u = Table[Floor[n*r], {n, 1, h}] (* A022839 *)
v = Table[Floor[n*s], {n, 1, h}] (* A108598 *)
w = Mod[u, 2] + Mod[v, 2] (* A191340 *)
Flatten[Position[w, 0]]  (* A191380 *)
Flatten[Position[w, 1]]  (* A191381 *)
Flatten[Position[w, 2]]  (* A191382 *)

a(n)=([nr] mod 2)+([ns] mod 2), where r=sqrt(5), s=r/(r-1), [ ]=floor.

A191333 Decimal representation of sum{(1/3)^A191331(k): k>=1}.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, May 31 2011

See A191335, A191331, and A191332.

			0.11248476074367570863455686829448049346...

Cf. A191329, A191331, A191332.

Mathematica
```
(See A191329.)
```

sum(k=1,2,(1/3.)^(1.<<(2^(2^k-1))+1)) \\ Charles R Greathouse IV, Jun 30 2011

A191335 Decimal expansion of sum{(1/3)^A005652(k): k>=1}.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 31 2011

See A191329 and A191334.

			0.3719512196066283453070417349013123446242...

Cf. A191329, A191334, A005652.

Mathematica
```
(See A191329.)
```

A191334 Decimal expansion of sum{(1/3)^A005653(k): k>=1}.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 31 2011

sum{(1/3)^A005653(k)}+sum{(1/3)^A005652(k)}=1/2.

			0.128048780393371654692958265098687655...

Cf. A191329, A191335.

Mathematica
```
(See A191329.)
```

A191339 Positions of 2 in A191336.

Original entry on oeis.org

3, 9, 10, 15, 16, 25, 26, 31, 32, 38, 41, 47, 48, 53, 54, 60, 63, 69, 70, 75, 76, 85, 91, 92, 98, 107, 108, 113, 114, 120, 123, 129, 130, 135, 136, 142, 145, 151, 152, 157, 158, 167, 173, 174, 180, 189, 190, 195, 196, 202, 205, 211, 212, 217, 218, 224, 227, 233, 234, 239, 240, 249, 250, 255, 256, 262, 271, 272, 277, 278, 284, 287

Offset: 1

Clark Kimberling, Jun 01 2011

nonn

See A191336.

Cf. A191339, A191329.

Mathematica
```
(See A191339.)
```

A191337 Positions of 0 in A191336.

Original entry on oeis.org

6, 7, 12, 13, 19, 22, 28, 29, 34, 35, 44, 50, 51, 56, 57, 66, 67, 72, 73, 79, 82, 88, 89, 94, 95, 101, 104, 110, 111, 116, 117, 126, 132, 133, 139, 148, 149, 154, 155, 161, 164, 170, 171, 176, 177, 183, 186, 192, 193, 198, 199, 208, 214, 215, 221, 230, 231, 236, 237, 243, 246, 252, 253, 258, 259, 265, 268, 274, 275, 280, 281, 290

Offset: 1

Clark Kimberling, Jun 01 2011

nonn

See A191336.

Cf. A191336, A191329.

Mathematica
```
(See A191336.)
```

cp's OEIS Frontend

A191330 Positions of 0 in A191329.

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A191331 Positions of 2 in A191329.

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A191336 (A022838 mod 2)+(A054406 mod 2).

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A191332 Decimal expansion of sum{(1/3)^A191330(k): k>=1}.

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A191340 (A022839 mod 2)+(A108598 mod 2).

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A191333 Decimal representation of sum{(1/3)^A191331(k): k>=1}.

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PARI

A191335 Decimal expansion of sum{(1/3)^A005652(k): k>=1}.

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A191334 Decimal expansion of sum{(1/3)^A005653(k): k>=1}.

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A191339 Positions of 2 in A191336.

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