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 1-3 of 3 results.

A211027 Triangle of binary numbers >= 1 with no initial repeats.

Original entry on oeis.org

1, 10, 100, 101, 1000, 1001, 1011, 10000, 10001, 10010, 10011, 10110, 10111, 100000, 100001, 100010, 100011, 100101, 100110, 100111, 101100, 101110, 101111, 1000000, 1000001, 1000010, 1000011, 1000100, 1000101, 1000110, 1000111, 1001010, 1001011, 1001100, 1001101, 1001110, 1001111, 1011000, 1011001, 1011100, 1011101, 1011110, 1011111
Offset: 1

Views

Author

Omar E. Pol, Nov 30 2012

Keywords

Comments

Triangle read by rows in which row n lists the binary numbers with n digits and with no initial repeats.
Also triangle read by rows in which row n lists the binary words of length n with no initial repeats and with initial digit 1. See also A211029.

Examples

			Triangle begins:
1;
10;
100, 101;
1000, 1001, 1011;
10000, 10001, 10010, 10011, 10110, 10111;
100000, 100001, 100010, 100011, 100101, 100110, 100111, 101100, 101110, 101111;
1000000, 1000001, 1000010, 1000011, 1000100, 1000101, 1000110, 1000111, 1001010, 1001011, 1001100, 1001101, 1001110, 1001111, 1011000, 1011001, 1011100, 1011101, 1011110, 1011111;

Links

Crossrefs

Column 1 is A011557. Row n has length A093371(n).
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-> 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
  • Mathematica
    T[n_] := If[n == 1, {1}, FromDigits /@ Select[Range[2^(n-1), 2^n-2] // IntegerDigits[#, 2]&, FindTransientRepeat[Reverse[#], 2][[2]] == {}&]];
Array[T, 7] // Flatten (* Jean-François Alcover, Feb 27 2021 *)

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 *)

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
Showing 1-3 of 3 results.

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