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.

A071055 Number of 0's in n-th row of triangle in A071038.

Original entry on oeis.org

0, 0, 2, 0, 2, 2, 6, 0, 2, 2, 6, 2, 6, 6, 14, 0, 2, 2, 6, 2, 6, 6, 14, 2, 6, 6, 14, 6, 14, 14, 30, 0, 2, 2, 6, 2, 6, 6, 14, 2, 6, 6, 14, 6, 14, 14, 30, 2, 6, 6, 14, 6, 14, 14, 30, 6, 14, 14, 30, 14, 30, 30, 62, 0, 2, 2, 6, 2, 6, 6, 14, 2, 6, 6, 14, 6, 14, 14, 30, 2, 6, 6, 14, 6
Offset: 0

Views

Author

Hans Havermann, May 26 2002

Keywords

Comments

a(n) is also the number of pairs of consecutive entries in the n-th row of Pascal's triangle with opposite parity.
All terms appear to be of the form 2^k - 2 (checked for n <= 10000). - Michael De Vlieger, Mar 02 2015
This appears to be equal to the number of previous values k, from 1..n-1, such that k AND n = k, where 'AND' is binary AND, and where the sequence starts at 1. For example, 1 AND 2 = 0, so a(2) = 0, while 1 AND 3 = 1 and 2 AND 3 = 2, so a(3) = 2. It follows from this that if n = 2^m - 1 then a(n) = n - 1 = 2^m - 2, giving the right border values noted below. - Scott R. Shannon, Apr 19 2023

Examples

			From _Omar E. Pol_, Mar 02 2015: (Start)
Also, written as an irregular triangle in which the row lengths are the powers of 2, the sequence begins:
0;
0,2;
0,2,2,6;
0,2,2,6,2,6,6,14;
0,2,2,6,2,6,6,14,2,6,6,14,6,14,14,30;
0,2,2,6,2,6,6,14,2,6,6,14,6,14,14,30,2,6,6,14,6,14,14,30,6,14,14,30,14,30,30,62;
...
It appears that the right border gives the nonnegative terms of A000918.
It appears that the row sums give A056182.
(End)

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.

Links

Crossrefs

Cf. A071042.

Programs

Formula

a(n)=b(n+1), with b(0)=0, b(2n)=b(n), b(2n+1)=2b(n)+2-2[n==0] (conjectured). - Ralf Stephan, Mar 05 2004
a(n) = pext(n, n + 1) (conjectured) where pext is the "parallel bits extract" instruction of the x86 CPU; pext(x, mask) extracts bits from x at the bit locations specified by mask to contiguous low bits. - Falk Hüffner, Jul 26 2019

A267610 Total number of OFF (white) cells after n iterations of the "Rule 182" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

0, 0, 2, 2, 4, 6, 12, 12, 14, 16, 22, 24, 30, 36, 50, 50, 52, 54, 60, 62, 68, 74, 88, 90, 96, 102, 116, 122, 136, 150, 180, 180, 182, 184, 190, 192, 198, 204, 218, 220, 226, 232, 246, 252, 266, 280, 310, 312, 318, 324, 338, 344, 358, 372, 402, 408, 422, 436
Offset: 0

Views

Author

Robert Price, Jan 18 2016

Keywords

Comments

From Gus Wiseman, Mar 30 2019: (Start)
It appears that a(n) is also the number of increasing binary-containment pairs of distinct positive integers up to n + 1. A pair of positive integers is a binary containment if the positions of 1's in the reversed binary expansion of the first are a subset of the positions of 1's in the reversed binary expansion of the second. For example, the a(2) = 2 through a(8) = 14 pairs are:
{1,3} {1,3} {1,3} {1,3} {1,3} {1,3} {1,3}
{2,3} {2,3} {1,5} {1,5} {1,5} {1,5} {1,5}
{2,3} {2,3} {1,7} {1,7} {1,7}
{4,5} {2,6} {2,3} {2,3} {1,9}
{4,5} {2,6} {2,6} {2,3}
{4,6} {2,7} {2,7} {2,6}
{3,7} {3,7} {2,7}
{4,5} {4,5} {3,7}
{4,6} {4,6} {4,5}
{4,7} {4,7} {4,6}
{5,7} {5,7} {4,7}
{6,7} {6,7} {5,7}
{6,7}
{8,9}
(End)

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

Crossrefs

Cf. A000120, A006046, A006218, A070939, A071038, A080572, A267700.
Cf. A325101, A325103, A325104, A325106, A325109, A325110, A325123, A325124.

Programs

  • Mathematica
    rule=182; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) nbc=Table[Total[catri[[k]]],{k,1,rows}]; (* Number of Black cells in stage n *) nwc=Table[Length[catri[[k]]]-nbc[[k]],{k,1,rows}]; (* Number of White cells in stage n *) Table[Total[Take[nwc,k]],{k,1,rows}] (* Number of White cells through stage n *)

Formula

Conjecture: a(n) = A267700(n) - n. - Gus Wiseman, Mar 30 2019
G.f.: (1/x)*(A(x)/x - (x+1)/(1-x)^2) where A(x) is the g.f. for A006046 (conjectured). - John Tyler Rascoe, Jul 08 2024

A071042 Number of 0's in n-th row of triangle in A070886.

Original entry on oeis.org

0, 1, 3, 3, 7, 7, 9, 7, 15, 15, 17, 15, 21, 19, 21, 15, 31, 31, 33, 31, 37, 35, 37, 31, 45, 43, 45, 39, 49, 43, 45, 31, 63, 63, 65, 63, 69, 67, 69, 63, 77, 75, 77, 71, 81, 75, 77, 63, 93, 91, 93, 87, 97, 91, 93, 79, 105, 99, 101, 87, 105, 91, 93, 63, 127, 127, 129, 127
Offset: 0

Views

Author

Hans Havermann, May 26 2002

Keywords

Comments

Also (omitting initial 0) number of 1's in n-th row of triangle in A071038, that is, number of ON cells at generation n of CA defined by Rule 182.

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.

Links

Crossrefs

Cf. A001316, A048967.

Programs

  • Mathematica
    Map[Function[Apply[Plus, Flatten[#1]]],
CellularAutomaton[182, {{1}, 0}, 100]] (* N. J. A. Sloane, Feb 17 2015 *)
  • PARI
    a(n)=2*n-2^hammingweight(n)+1; \\ Altug Alkan, Dec 05 2015

Formula

a(n) = 2n + 1 - A001316(n) = n + A048967(n).
a(2n) = a(n) + 2n, a(2n+1) = 2a(n) + 1. - Ralf Stephan, Oct 07 2003

A267608 Binary representation of the n-th iteration of the "Rule 182" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 111, 10101, 1111111, 101111101, 11101110111, 1010101010101, 111111111111111, 10111111111111101, 1110111111111110111, 101010111111111010101, 11111110111111101111111, 1011111010111110101111101, 111011101110111011101110111, 10101010101010101010101010101
Offset: 0

Views

Author

Robert Price, Jan 18 2016

Keywords

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

Crossrefs

Cf. A071038.

Programs

  • Mathematica
    rule=182; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]]],{k,1,rows}]   (* Binary Representation of Rows *)

A267609 Decimal representation of the n-th iteration of the "Rule 182" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 7, 21, 127, 381, 1911, 5461, 32767, 98301, 491511, 1408981, 8355711, 25001341, 125269879, 357913941, 2147483647, 6442450941, 32212254711, 92341796821, 547608330111, 1638530022781, 8209829984119, 23456963876181, 140735340838911, 422201727483901
Offset: 0

Views

Author

Robert Price, Jan 18 2016

Keywords

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

Crossrefs

Cf. A071038.

Programs

  • Mathematica
    rule=182; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]],2],{k,1,rows}]   (* Decimal Representation of Rows *)
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.