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.

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A191323 Increasing sequence generated by these rules: a(1)=1, and if x is in a then [3x/2]+1 and 3x+1 are in a, where [ ]=floor.

Original entry on oeis.org

1, 2, 4, 7, 11, 13, 17, 20, 22, 26, 31, 34, 40, 47, 52, 61, 67, 71, 79, 92, 94, 101, 103, 107, 119, 121, 139, 142, 152, 155, 157, 161, 179, 182, 184, 202, 209, 214, 229, 233, 236, 238, 242, 269, 274, 277, 283, 304, 310, 314, 322, 344, 350, 355, 358, 364, 404, 412, 416, 418, 425, 427, 457, 466, 472, 484, 517, 526, 533, 538, 547, 553
Offset: 1

Views

Author

Clark Kimberling, May 30 2011

Keywords

Comments

This sequence represents a class of sequences generated by rules of the form "a(1)=1, and if x is in a then floor(hx+i) and floor(jx+k) are in a, where h and j are rational numbers and i and k are positive integers." In the following examples, the floor function is denoted by [ ].
A191323: [3x/2]+1, 3x+1
A191324: [3x/2]+1, 3x+2
A191325: [3x/2], [5x/2]
A191326: [3x/2], [7x/2]
A191327: [5x/2], [7x/2]
A191328: [5x/3], [7x/3]
Other families of sequences generated by "rules" are listed at A191803, A191106, A101113 and A191203.

Examples

			1 -> 2,4 -> 6,7,13 -> 10,11,19,20,22,40 -> ...

Links

Crossrefs

Cf. A190503, A191100, A191113, A191203.

Programs

  • Mathematica
    h = 3; i = 1; j = 3; k = 1; f = 1; g = 12;
a=Union[Flatten[NestList[{Floor[h#/2]+i,j#+k}&,f,g]]]
(* A191323 *)

A173934 Irregular triangle in which row n consists of numbers m < k/2 such that m/k is in the Cantor set, where k= A173931(n) and gcd(m,k) = 1.

Original entry on oeis.org

1, 1, 3, 1, 3, 4, 1, 3, 9, 1, 3, 9, 13, 1, 3, 7, 9, 19, 21, 25, 27, 1, 3, 9, 10, 27, 30, 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37, 39, 40, 5, 11, 15, 33, 45, 47, 5, 15, 41, 45, 47, 59, 7, 16, 21, 22, 48, 61, 63, 66, 1, 3, 7, 9, 19, 21, 25, 27, 55, 57, 63, 73, 75, 79, 81, 1, 3, 9, 27
Offset: 1

Views

Author

T. D. Noe, Mar 03 2010

Keywords

Comments

The length of row n is A173933(n). Observe that the m are actually less than k/3. Note that (k-m)/k is also in the Cantor set. If m appears in a row, then 3m does also. Let A and B be the first and last numbers in row n, then it appears that k = A + 3B. This implies A = k (mod 3). The interesting graph of this triangle shows that some ranges of m are not allowed.
When k is a prime of the form (3^r-1)/2, then the row consists of the 2^(r-1)-1 numbers (greater than 0) whose base-3 representation consists of only 0's and 1's. Hence, for r=3,7, and 13, the primes k are 13, 1093, and 797161, and the number of m < k/2 is 3, 63, and 4095.

Links

Crossrefs

Cf. A005836, A007734, A054591, A173931, A173933, A191106, A306556.

Programs

  • Mathematica
    Flatten[Last[Transpose[cantor]]] (* see A173931 *)

Extensions

Name qualified by Peter Munn, Jul 06 2019

A191109 a(1)=1, and if x is a term then 3x-1 and 3x+2 are terms too.

Original entry on oeis.org

1, 2, 5, 8, 14, 17, 23, 26, 41, 44, 50, 53, 68, 71, 77, 80, 122, 125, 131, 134, 149, 152, 158, 161, 203, 206, 212, 215, 230, 233, 239, 242, 365, 368, 374, 377, 392, 395, 401, 404, 446, 449, 455, 458, 473, 476, 482, 485, 608, 611, 617, 620, 635, 638, 644, 647, 689, 692, 698, 701, 716, 719, 725, 728, 1094, 1097, 1103, 1106, 1121
Offset: 1

Views

Author

Clark Kimberling, May 26 2011

Keywords

Comments

See discussions at A190803, A191106.
The positive integers in (1+A191109)/3 comprise A153775, a proper subsequence of A191109.
The positive integers in (-2+A191109)/3 comprise A032924, a proper subsequence of A191109.

Links

Crossrefs

Cf. A190803, A191106, A153775, A032924.

Programs

  • Mathematica
    h = 3; i = -1; j = 3; k = 2; f = 1;  g = 7;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* 191109 *)
b = (a + 1)/3; c = (a - 2)/3; r = Range[1, 900];
d = Intersection[b, r] (* A153775 *)
e = Intersection[c, r] (* A032924 *)
Nest[Flatten[{#,3#-1,3#+2}]&,1,10]//Union (* Harvey P. Dale, Apr 05 2020 *)

Extensions

Name edited by Michel Marcus, Jul 29 2021

A191110 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x and 3x+2 are in a.

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 17, 27, 29, 33, 35, 45, 47, 51, 53, 81, 83, 87, 89, 99, 101, 105, 107, 135, 137, 141, 143, 153, 155, 159, 161, 243, 245, 249, 251, 261, 263, 267, 269, 297, 299, 303, 305, 315, 317, 321, 323, 405, 407, 411, 413, 423, 425, 429, 431, 459, 461, 465, 467, 477, 479, 483, 485, 729, 731, 735, 737, 747, 749, 753, 755, 783
Offset: 1

Views

Author

Clark Kimberling, May 26 2011

Keywords

Comments

See discussions at A190803, A191106. A191110 has closure properties: the positive integers in (A191110)/3 form A191110, and likewise for (-2+A191110).

Links

Crossrefs

Cf. A190803, A191106.

Programs

  • Mathematica
    h = 3; i = 0; j = 3; k = 2; f = 1;  g = 7;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A191110 *)
b = a/3; c = (a - 2)/3; r = Range[0, 900];
d = Intersection[b, r] (* A191110 closure property  *)
e = Intersection[c, r] (* A191110 closure property  *)
Flatten[Nest[{#,3#,3#+2}&/@#&,{1},6]]//Union (* Harvey P. Dale, Sep 30 2019 *)
Previous Showing 11-14 of 14 results.

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