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 1-8 of 8 results.

A057986 Positions of 0 in A057985.

Original entry on oeis.org

1, 7, 10, 11, 15, 16, 18, 24, 25, 27, 31, 37, 40, 41, 43, 47, 53, 54, 60, 63, 64, 68, 69, 71, 75, 81, 82, 88, 91, 92, 94, 100, 103, 104, 108, 109, 111, 117, 118, 120, 124, 130, 131, 137, 140, 141, 143, 149, 152, 153, 157, 158, 160, 164
Offset: 1

Views

Author

Clark Kimberling, Oct 30 2000

Keywords

Links

Crossrefs

Cf. A057985, A057987, A057988.

A057987 Positions of 1 in A057985.

Original entry on oeis.org

2, 3, 5, 8, 12, 13, 17, 19, 20, 22, 26, 28, 29, 32, 33, 35, 38, 42, 44, 45, 48, 49, 51, 55, 56, 58, 61, 65, 66, 70, 72, 73, 76, 77, 79, 83, 84, 86, 89, 93, 95, 96, 98, 101, 105, 106, 110, 112, 113, 115, 119, 121, 122, 125, 126, 128, 132, 133
Offset: 1

Views

Author

Clark Kimberling, Oct 30 2000

Keywords

Links

Crossrefs

Cf. A057985, A057986, A057988.

A057988 Positions of 2 in A057985.

Original entry on oeis.org

4, 6, 9, 14, 21, 23, 30, 34, 36, 39, 46, 50, 52, 57, 59, 62, 67, 74, 78, 80, 85, 87, 90, 97, 99, 102, 107, 114, 116, 123, 127, 129, 134, 136, 139, 146, 148, 151, 156, 163, 167, 169, 172, 177, 184, 186, 193, 197, 199, 202, 209, 213
Offset: 1

Views

Author

Clark Kimberling, Oct 30 2000

Keywords

Links

Crossrefs

Cf. A057985, A057986, A057987.

A057989 Positions of 00 or 11 or 22 in A057985.

Original entry on oeis.org

2, 10, 12, 15, 19, 24, 28, 32, 40, 44, 48, 53, 55, 63, 65, 68, 72, 76, 81, 83, 91, 95, 103, 105, 108, 112, 117, 121, 125, 130, 132, 140, 144, 152, 154, 157, 161, 165, 173, 175, 178, 182, 187, 191, 195, 203, 207, 211, 216, 218, 226, 230
Offset: 1

Views

Author

Clark Kimberling, Oct 30 2000

Keywords

A057990 Positions of 00 in A057985.

Original entry on oeis.org

10, 15, 24, 40, 53, 63, 68, 81, 91, 103, 108, 117, 130, 140, 152, 157, 173, 178, 187, 203, 216, 226, 238, 243, 259, 264, 273, 286, 296, 301, 310, 326, 339, 349, 354, 367, 377, 389, 394, 410, 415, 424, 437, 447, 452, 461, 477, 490
Offset: 1

Views

Author

Clark Kimberling, Oct 30 2000

Keywords

A287072 Start with 0 and repeatedly substitute 0->01, 1->21, 2->0.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 21 2017

Keywords

Comments

A fixed point of the morphism 0->01, 1->21, 2->0. Let u be the sequence of positions of 0, and likewise, v for 1 and w for 2. Let U, V, W be the limits of u(n)/n, v(n)/n, w(n)/n, respectively. Then 1/U + 1/V + 1/W = 1, where
U = 3.079595623491438786010417...,
V = 2.324717957244746025960908...,
W = U + 1 = 4.079595623491438786010417....
Since the morphism 0->01, 1->21, 2->0 is the time reversal of the morphism 0->10, 1->12 2->0, which has fixed point A287104, in particular the incidence matrices of these two morphisms are equal. Thus the algebraic expressions found for U, V and W in A287104 do also apply to the U, V and W above. - Michel Dekking, Sep 15 2019
If n >=2, then u(n) - u(n-1) is in {2,3,4}, v(n) - v(n-1) is in {2,3}, and w(n) - w(n-1) is in {3,4,5}.

Links

Crossrefs

Cf. A057985, A287073, A287074, A287075, A287104.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {2, 1}, 2 -> 0}] &, {0}, 10] (* A287072 *)
Flatten[Position[s, 0]] (* A287073 *)
Flatten[Position[s, 1]] (* A287074 *)
Flatten[Position[s, 2]] (* A287075 *)
SubstitutionSystem[{0->{0,1},1->{2,1},2->{0}},{0},{8}][[1]] (* Harvey P. Dale, Feb 18 2025 *)

A287066 Start with 1 and repeatedly substitute 0->01, 1->12, 2->0.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 20 2017

Keywords

Comments

This is the fixed point of the morphism 0->01, 1->12, 2->0 starting with 1. Let u be the sequence of positions of 0, and likewise, v for 1 and w for 2. Let U, V, W be the limits of u(n)/n, v(n)/n, w(n)/n, respectively. Then 1/U + 1/V + 1/W = 1, where
U = 3.079595623491438786010417...,
V = 2.324717957244746025960908...,
W = U + 1 = 4.079595623491438786010417....
If n >=2, then u(n) - u(n-1) is in {1,2,3,4,6}, v(n) - v(n-1) is in {1,2,3,4}, and w(n) - w(n-1) is in {2,3,4,5,7}. For n >= 1, the number of terms resulting from n iterations of the morphism is A005251(n+2).

Links

Crossrefs

Cf. A057985, A287067, A287068, A287069.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 2}, 2 -> 0}] &, {1}, 10] (* A287066 *)
Flatten[Position[s, 0]]  (* A287067 *)
Flatten[Position[s, 1]]  (* A287068 *)
Flatten[Position[s, 2]]  (* A287069 *)

A248335 A recursive sequence generated by an L-system defined in comments.

Original entry on oeis.org

1, 23, 3445, 45565667, 5667677867787889, 6778788978898990788989908990901, 7889899089909018990901901112899090190111290111211223, 89909019011129011121122390111211223112232323349011121122311223232334112232323342323343445
Offset: 1

Views

Author

Felix Fröhlich, Oct 26 2014

Keywords

Comments

The L-system producing the sequence is defined as follows:
Alphabet: 1 2 3 4 5 6 7 8 9 0
Initiator: 1
Production rules: (1 --> 23), (2 --> 34), (3 --> 45), (4 --> 56), (5 --> 67), (6 --> 78), (7 --> 89), (8 --> 90), (9 --> 1), (0 --> 12).

Links

Crossrefs

Cf. A057985, A073058, A103684.
Showing 1-8 of 8 results.

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