cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 19 results. Next

A191251 Positions of 0 in A191250.

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 27, 29, 31, 33, 35, 37, 39, 40, 42, 44, 46, 48, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 68, 70, 72, 74, 76, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 115, 117, 119, 121, 123, 125, 127, 128, 130, 132, 134, 136, 137, 139, 141, 143, 145, 147, 149
Offset: 1

Views

Author

Clark Kimberling, May 28 2011

Keywords

Comments

See A191250.

Crossrefs

Cf. A191250.

Programs

A191252 Positions of 1 in A191250.

Original entry on oeis.org

2, 6, 9, 11, 15, 17, 21, 25, 28, 30, 34, 38, 41, 43, 47, 50, 52, 56, 58, 62, 66, 69, 71, 75, 78, 80, 84, 86, 90, 94, 97, 99, 103, 105, 109, 113, 116, 118, 122, 126, 129, 131, 135, 138, 140, 144, 146, 150, 154, 157, 159, 163, 165, 169, 173, 176, 178, 182, 186, 189, 191, 195, 198, 200, 204, 206, 210, 214, 217
Offset: 1

Views

Author

Clark Kimberling, May 28 2011

Keywords

Comments

See A191250.

Crossrefs

Cf. A191250.

Programs

A191253 Positions of 2 in A191250.

Original entry on oeis.org

4, 13, 19, 23, 32, 36, 45, 54, 60, 64, 73, 82, 88, 92, 101, 107, 111, 120, 124, 133, 142, 148, 152, 161, 167, 171, 180, 184, 193, 202, 208, 212, 221, 225, 234, 243, 249, 253, 262, 271, 277, 281, 290, 296, 300, 309, 313, 322, 331, 337, 341, 350, 354, 363, 372, 378, 382, 391, 400, 406
Offset: 1

Views

Author

Clark Kimberling, May 28 2011

Keywords

Comments

See A191250.

Crossrefs

Cf. A191250.

Programs

A191255 Fixed point of the morphism 0 -> 01, 1 -> 02, 2 -> 03, 3 -> 01.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2
Offset: 1

Views

Author

Clark Kimberling, May 28 2011

Keywords

Comments

See A191250.
The asymptotic density of the occurrences of k = 0, 1, 2 and 3 is 1/2, 2/7, 1/7 and 1/14, respectively. The asymptotic mean of this sequence is 11/14. - Amiram Eldar, May 31 2024

Links

Crossrefs

Cf. A191250, A191254, A373157.
Positions of 0 or 3: A191257; positions of 0: A005408; positions of 1: A067368; positions of 2: A213258.

Programs

  • Mathematica
    t = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 2}, 2 -> {0, 3}, 3 -> {0, 1}}] &, {0}, 9] (* this sequence *)
Flatten[Position[t, 0]] (* A005408, the odds *)
a = Flatten[Position[t, 1]] (* A067368 *)
b = Flatten[Position[t, 2]] (* A213258 *)
a/2  (* A191257 *)
b/4  (* a/2 *)
  • PARI
    A191255(n) = if(n%2, 0, my(e=valuation(n, 2)%3); if(!e, 3, e)); \\ Antti Karttunen, May 28 2024, after Jianing Song's Sep 21 2018 formula

Formula

a(n) = 0 for odd n, otherwise a(n) is the unique number in {1,2,3} that is congruent to v2(n) modulo 3, where v2(n) = A007814(n) is the 2-adic valuation of n. - Jianing Song, Sep 21 2018 [Clarified by Jianing Song, May 30 2024]
Recurrence: a(2n-1) = 0, a(2n) = 1, 2, 3, 1 for a(n) = 0, 1, 2, 3 respectively. - Jianing Song, May 30 2024

A191261 Fixed point of the morphism 0 -> 01, 1 -> 002, 2 -> 01.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 0, 2, 0, 1, 0, 0, 2, 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 0, 2, 0, 1, 0, 0, 2, 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 0, 2, 0, 1, 0, 0, 2, 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 0, 2
Offset: 1

Views

Author

Clark Kimberling, May 28 2011

Keywords

Comments

See A191250.

Crossrefs

Cf. A191250, A191262, A191263, A191264.

Programs

  • Mathematica
    t = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 0, 2}, 2 -> {0, 1}}] &, {0}, 8]  (* A191261 *)
Flatten[Position[t, 0]] (* A191262 *)
Flatten[Position[t, 1]] (* A191263 *)
Flatten[Position[t, 2]] (* A191264 *)

A191265 Fixed point of the morphism 0 -> 001, 1 -> 002, 2 -> 01.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0
Offset: 1

Views

Author

Clark Kimberling, May 28 2011

Keywords

Comments

See A191265.

Crossrefs

Cf. A191265, A191250.

Programs

  • Mathematica
    t = Nest[Flatten[# /. {0 -> {0, 0, 1}, 1 -> {0, 0, 2}, 2 -> {0, 1}}] &, {0}, 7] (* A191265 *)
Flatten[Position[t, 0]]  (* A191266 *)
Flatten[Position[t, 1]]  (* A191267 *)
Flatten[Position[t, 2]]  (* A191268 *)

A191258 Fixed point of the morphism 0->01, 1->02, 2->03, 3->001.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3
Offset: 1

Views

Author

Clark Kimberling, May 28 2011

Keywords

Comments

See A191250.

Crossrefs

Cf. A191250, A191259, A191260.

Programs

  • Mathematica
    t = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 2}, 2 -> {0, 3}, 3 -> {0, 0, 1}}] &, {0}, 10]
(* A191258 *)
Flatten[Position[t, 0]] (* A005408, the odds *)
Flatten[Position[t, 1]] (* A191259 *)
Flatten[Position[t, 2]] (* A191260 *)
Flatten[SubstitutionSystem[{0->{0,1},1->{0,2},2->{0,3},3->{0,0,1}},{0},{10}]] (* Harvey P. Dale, Nov 27 2021 *)

A191263 Positions of 1 in A191261.

Original entry on oeis.org

2, 7, 9, 11, 13, 18, 23, 28, 33, 35, 37, 39, 44, 46, 48, 50, 55, 57, 59, 61, 66, 68, 70, 72, 77, 82, 87, 92, 94, 96, 98, 103, 108, 113, 118, 120, 122, 124, 129, 134, 139, 144, 146, 148, 150, 155, 160, 165, 170, 172, 174, 176, 181, 183, 185, 187, 192, 194, 196, 198, 203, 205, 207, 209, 214, 219, 224, 229, 231, 233, 235, 240, 242, 244
Offset: 1

Views

Author

Clark Kimberling, May 28 2011

Keywords

Comments

Also, positions of 1 in the fixed point of the morphism 0->01, 1>000; see A284745. - Clark Kimberling, Apr 13 2017

Links

Crossrefs

Cf. A191261, A191250, A284745, A284746.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 0, 0}}] &, {0}, 13]; (* A284745 *)
Flatten[Position[s, 0]];  (* A284746 *)
Flatten[Position[s, 1]];  (* A191263 *)

A191269 Fixed point of the morphism 0 -> 001, 1 -> 02, 2 -> 01.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 1, 0
Offset: 1

Views

Author

Clark Kimberling, May 28 2011

Keywords

Comments

See A191250.
Proof of Kimberling's conjecture on the positions of 0 in this sequence: consider the letter to letter projection pi given by pi(0) = 0, pi(1) = 1, pi(2) = 1. Then pi sigma = tau pi, where tau is the morphism on {0,1} given by tau(0) = 001, tau(1) = 01. It follows that pi(a) = x, where x = A188432 is the fixed point of tau. Note that the positions of zero in a = A191269 are equal to the positions of zero in x. Since x is the infinite Fibonacci word with a zero in front, it follows that these positions are given by A026351. - Michel Dekking, Aug 24 2019

Crossrefs

Cf. A191250, A191270, A191271.

Programs

  • Mathematica
    t = Nest[Flatten[# /. {0 -> {0, 0, 1}, 1 -> {0, 2}, 2 -> {0, 1}}] &, {0}, 7] (* A191269 *)
Flatten[Position[t, 0]]  (* A026351, 1+lower Wythoff sequence, conjectured *)
Flatten[Position[t, 1]] (* A191270 *)
Flatten[Position[t, 2]] (* A191271 *)

A191254 Fixed point of the morphism 0 -> 01, 1 -> 02, 2 -> 01.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2
Offset: 1

Views

Author

Clark Kimberling, May 28 2011

Keywords

Comments

For related sequences, see notes in the Mathematica program.
The asymptotic density of the occurrences of k = 0, 1 and 2 is 1/2, 1/3 and 1/6, respectively. The asymptotic mean of this sequence is 2/3. - Amiram Eldar, May 31 2024

Links

Crossrefs

Cf. A191250, A191255.
Positions of 0 or 2: A003159; positions of 0: A005408; positions of 1: A036554; positions of 2: A108269.

Programs

  • Mathematica
    t = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 2}, 2 -> {0, 1}}] &, {0}, 9]  (* A191254 *)
Flatten[Position[t, 0]]  (* A005408, the odds *)
a = Flatten[Position[t, 1]] (* A036554 *)
b = Flatten[Position[t, 2]] (* A108269 *)
a/2 (* A003159 *)
b/4 (* A003159 *)
  • PARI
    A191254(n) = if(n%2,0,if(valuation(n,2)%2,1,2)); \\ Antti Karttunen, Nov 06 2018

Formula

From Jianing Song, May 30 2024: (Start)
Recurrence: a(2n-1) = 0, a(2n) = 1, 2, 1 for a(n) = 0, 1, 2 respectively.
a(n) = 0 for odd n; a(n) = 1 for even n such that v2(n) is odd; a(n) = 2 for even n such that v2(n) is even, where v2(n) = A007814(n) is the 2-adic valuation of n. (End)
Showing 1-10 of 19 results. Next

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