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 1-5 of 5 results.

A211967 Triangle of decimal equivalents of binary numbers with no initial repeats, A211027.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 11, 16, 17, 18, 19, 22, 23, 32, 33, 34, 35, 37, 38, 39, 44, 46, 47, 64, 65, 66, 67, 68, 69, 70, 71, 74, 75, 76, 77, 78, 79, 88, 89, 92, 93, 94, 95, 128, 129, 130, 131, 132, 133, 134, 135, 137, 138, 139, 140, 141, 142, 143, 148, 149, 150
Offset: 1

Views

Author

Omar E. Pol, Nov 30 2012

Keywords

Examples

			Irregular triangle begins:
1;
2;
4,   5;
8,   9, 11;
16, 17, 18, 19, 22, 23;
32, 33, 34, 35, 37, 38, 39, 44, 46, 47;

Links

Crossrefs

Columns 1-2 give: A000079(n-1), A000051(n-1) for n>2. Row n has length A093371(n). Right border gives A083329(n-1).
Cf. A122536, A211029, A211968, A211969, A216955.

Programs

  • Maple
    s:= proc(n) s(n):= `if`(n=1, [[1]], map(x->
      [[x[], 0], [x[], 1]][], s(n-1))) end:
T:= proc(n) map (x-> add(x[i]*2^(nops(x)-i), i=1..nops(x)), select
      (proc(l) local i; for i to iquo(nops(l), 2) do if l[1..i]=
      l[i+1..2*i] then return false fi od; true end, s(n)))[] end:
seq (T(n), n=1..8);  # Alois P. Heinz, Dec 03 2012

A211029 Triangle read by rows in which row n lists the binary words of length n over the alphabet {1,2} with no initial repeats.

Original entry on oeis.org

1, 2, 12, 21, 121, 122, 211, 212, 1211, 1221, 1222, 2111, 2112, 2122, 12111, 12112, 12211, 12212, 12221, 12222, 21111, 21112, 21121, 21122, 21221, 21222, 121111, 121112, 121122, 122111, 122112, 122121, 122211, 122212, 122221, 122222, 211111, 211112
Offset: 1

Views

Author

Omar E. Pol, Nov 29 2012

Keywords

Comments

As usual in the OEIS, binary alphabets are encoded with {1,2} over the alphabet {0,1} the entries contain nonzero "numbers" beginning with 0.

Examples

			The fourth row of triangle of binary sequences is
0100, 0110, 0111, 1000, 1001, 1011 (see section example of A122536) therefore the fourth row of this triangle is
1211, 1221, 1222, 2111, 2112, 2122.
The first six rows of triangle are:
1, 2;
12, 21;
121, 122, 211, 212;
1211, 1221, 1222, 2111, 2112, 2122;
12111, 12112, 12211, 12212, 12221, 12222, 21111, 21112, 21121, 21122, 21221, 21222;
121111, 121112, 121122, 122111, 122112, 122121, 122211, 122212, 122221, 122222, 211111, 211112, 211121, 211122, 211212, 211221, 211222, 212211, 212221, 212222;

Links

Crossrefs

Row n has length A122536(n).
Cf. A093371, A211027, A213969, A216955.

Programs

  • Maple
    s:= proc(n) s(n):= `if`(n=0, [[]], map(x->
      [[x[], 1], [x[], 2]][], s(n-1))) end:
T:= proc(n) map(x-> parse(cat(x[])), select(proc(l) local i;
      for i to iquo(nops(l), 2) do if l[1..i]=l[i+1..2*i]
      then return false fi od; true end, s(n)))[] end:
seq(T(n), n=1..7);  # Alois P. Heinz, Dec 02 2012

Extensions

More terms and name improved by R. J. Mathar, Nov 30 2012

A211968 Triangle of binary numbers with some initial repeats.

Original entry on oeis.org

11, 110, 111, 1010, 1100, 1101, 1110, 1111, 10100, 10101, 11000, 11001, 11010, 11011, 11100, 11101, 11110, 11111, 100100, 101000, 101001, 101010, 101011, 101101, 110000, 110001, 110010, 110011, 110100, 110101, 110110, 110111, 111000, 111001, 111010, 111011
Offset: 2

Views

Author

Omar E. Pol, Dec 03 2012

Keywords

Comments

Triangle read by rows in which row n lists the binary numbers with n digits and with some initial repeats, n >= 2.
Also triangle read by rows in which row n lists the binary words of length n with some initial repeats and with initial digit 1, n >= 2.

Examples

			Triangle begins, starting at row 2:
  11;
  110, 111;
  1010, 1100, 1101, 1110, 1111;
  10100, 10101, 11000, 11001, 11010, 11011, 11100, 11101, 11110, 11111;

Links

Crossrefs

Complement in base 2 of A211027.
Rows lengths give: A093370.
Cf. A093370, A093371, A121880, A122536, A211967, A211969, A211973, A216955.

Programs

  • Maple
    s:= proc(n) s(n):= `if`(n=1, [[1]], map(x->
      [[x[], 0], [x[], 1]][], s(n-1))) end:
T:= proc(n) map(x-> parse(cat(x[])), select(proc(l) local i;
      for i to iquo(nops(l), 2) do if l[1..i]=l[i+1..2*i]
      then return true fi od; false end, s(n)))[] end:
seq(T(n), n=2..7);  # Alois P. Heinz, Dec 04 2012
  • Mathematica
    T[n_] := FromDigits /@ Select[Range[2^(n-1), 2^n-1] // IntegerDigits[#, 2]&, FindTransientRepeat[Reverse[#], 2][[2]] != {}&];
Table[T[n], {n, 2, 7}] // Flatten (* Jean-François Alcover, Feb 12 2025 *)

A211969 Triangle of decimal equivalents of binary numbers with some initial repeats, A211968.

Original entry on oeis.org

3, 6, 7, 10, 12, 13, 14, 15, 20, 21, 24, 25, 26, 27, 28, 29, 30, 31, 36, 40, 41, 42, 43, 45, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 72, 73, 80, 81, 82, 83, 84, 85, 86, 87, 90, 91, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106
Offset: 2

Views

Author

Omar E. Pol, Dec 03 2012

Keywords

Examples

			Irregular triangle begins, starting at row 2:
3;
6, 7;
10, 12, 13, 14, 15;
20, 21, 24, 25, 26, 27, 28, 29, 30, 31;
36, 40, 41, 42, 43, 45, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63;

Links

Crossrefs

Complement of A211967.
Row lengths give: A093370.
Column 1 gives: A005418(n+1).
Right border gives: A000225(n).
Cf. A093370, A093371, A121880, A122536, A211027, A211029, A211973, A216955.

Programs

  • Maple
    s:= proc(n) s(n):= `if`(n=1, [[1]], map(x->
      [[x[], 0], [x[], 1]][], s(n-1))) end:
T:= proc(n) map (x-> add(x[i]*2^(nops(x)-i), i=1..nops(x)), select
      (proc(l) local i; for i to iquo(nops(l), 2) do if l[1..i]=
      l[i+1..2*i] then return true fi od; false end, s(n)))[] end:
seq (T(n), n=2..7);  # Alois P. Heinz, Dec 04 2012

A328072 List of Nyldon words over {0,1}.

Original entry on oeis.org

0, 1, 10, 100, 101, 1000, 1001, 1011, 10000, 10001, 10010, 10011, 10110, 10111, 100000, 100001, 100010, 100011, 100110, 100111, 101100, 101110, 101111, 1000000, 1000001, 1000010, 1000011, 1000100, 1000110, 1000111, 1001010, 1001100, 1001110, 1001111, 1011000, 1011001, 1011010, 1011100, 1011101, 1011110, 1011111
Offset: 1

Views

Author

N. J. A. Sloane, Oct 07 2019

Keywords

Links

  • Émilie Charlier, Manon Philibert, Manon Stipulanti, Nyldon words, arXiv:1804.09735 [math.CO], 2018. Also J. Comb. Thy. A, 167 (2019), 60-90. See Table 1.

Crossrefs

Cf. A001037, A102659, A328073.
Initially agrees with A211027 but that is a different sequences.
Showing 1-5 of 5 results.

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