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.

A306514 Decimal representation of binary numbers with string structure 10s00, s in {0,1}*, such that it results in a non-palindromic cycle of length 4 in the Reverse and Add! procedure in base 2.

Original entry on oeis.org

84, 180, 324, 360, 684, 744, 1416, 1488, 2628, 2904, 3024, 5580, 5904, 6048, 10836, 11400, 11952, 12192, 21060, 21684, 23220, 23448, 23556, 24096, 24384, 43188, 43668, 44604, 44748, 46248, 46260, 47376, 48480, 48960, 86388, 86964, 91272, 92520, 92532, 93108, 95592, 96048, 96264, 97344, 97920, 166212
Offset: 1

Views

Author

A.H.M. Smeets, Feb 21 2019

Keywords

Comments

If the decimal representation of the binary string 10s00 is in the sequence, so is 101s000.
For binary representation see A306515.
This sequence is a subset of A066059.
These regular patterns can be represented by the context-free grammar with production rules:
S -> S_a | S_b | S_c | S_d
S_a -> 10 T_a 00, T_a -> 1 T_a 0 | T_a0,
S_b -> 11 T_b 01, T_b -> 0 T_b 1 | T_b0,
S_c -> 10 T_c 000, T_c -> 1 T_c 0 | T_c0,
S_d -> 11 T_d 101, T_d -> 0 T_d 1 | T_d0,
where T_a0, T_b0, T_c0 and T_d0 are some terminating strings.
Numbers in this sequence are generated by choosing S_a or S_c from the starting symbol S.
The decimal representation of all binary numbers derived by S -> S_a -> 10 T_a 00 -> 10 T_a0 00 are given in sequence A306516, its binary representation in A306517.
Observed: all values are in the ranges lower(k)..upper(k), where lower(k) = 81*2^k + 2^floor((k+6)/2) + 2^6*(2^floor((k-8)/2) - 1) + 4, which holds for k >= 11, and upper(k) = 3*2^floor((k+4)/2)*(2^floor((k+7)/2) - 1), which holds for k >= 0; the number of terms in each successive range increases by about a factor of 4/3. All terms between lower(k) and upper(k) are represented by a (k+7)-binary-digit number (see A306515). Each m-binary-digit number will have a successive number of m+1 binary digits in the next range. About 1/4 of each obtained number in this sequence has a new unique cyclic trajectory (see A306516 and A306517), i.e., a cyclic trajectory not joining a previous cyclic trajectory, which explains the growth factor of 4/3 for each successive range.
All terms A061561(4k+2) for k >= 0 are included in this sequence.
All values in A103897(k+3) for k >= 0 are included in this sequence.

Examples

			a(45) = 97920 = upper(10)
The following burst of terms is from a(46) = 166212 = lower(11) up to and including a(60) = 196224 = upper(12).
The burst of terms corresponding with k = 28 is from lower(28) = 21743468484 = a(5276) up to and including upper(28) = 25769607168 = a(6976).

Links

Crossrefs

Cf. A061561, A103897, A306515, A306516, A306517.

Formula

a(n) = 0 (mod 12).

A195460 a(n) = 2^(2*n+1) - 3*2^n - 1.

Original entry on oeis.org

1, 19, 103, 463, 1951, 7999, 32383, 130303, 522751, 2094079, 8382463, 33542143, 134193151, 536821759, 2147385343, 8589737983, 34359345151, 137438167039, 549754241023, 2199020109823, 8796086730751, 35184359505919, 140737463189503
Offset: 1

Views

Author

Brad Clardy, Sep 18 2011

Keywords

Comments

Binary numbers of the form (n)00(m) where n and m are the number of 1's, m is the index and n=m-1.

Examples

			First few terms in binary are 1, 10011, 1100111, 111001111, 11110011111, 1111100111111.

Links

Crossrefs

Cf. A052539, A103897, A195461 (prime values).

Programs

  • Magma
    [2*4^n-3*2^n-1: n in [1..23]]; // Bruno Berselli, Sep 19 2011
  • PARI
    a(n)=1<<(2*n+1)-3<Charles R Greathouse IV, Sep 19 2011

Formula

a(n) = 2*a(n-1)+A052539(n) = a(n-1)+A103897(n) for n>1.
G.f.: x*(1+12*x-16*x^2)/((1-x)*(1-2*x)*(1-4*x)). - Bruno Berselli, Sep 19 2011

A346306 Position in A076478 of the binary complement of the n-th word in A076478.

Original entry on oeis.org

2, 1, 6, 5, 4, 3, 14, 13, 12, 11, 10, 9, 8, 7, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 126, 125, 124, 123
Offset: 1

Views

Author

Clark Kimberling, Aug 16 2021

Keywords

Comments

Permutation of the positive integers obtained by reversing their order within successive subsets of length 2, 4, 8, 16, ... - Paolo Xausa, Mar 09 2023

Examples

			The first fourteen words w(n) are 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, 110, 111, so that a(3) = 6.
From _Paolo Xausa_, Mar 09 2023: (Start)
Written as an irregular triangle, where row r >= 1 has length 2^r and row sum is A103897(r), the sequence begins:
   2,  1;
   6,  5,  4,  3;
  14, 13, 12, 11, 10,  9,  8,  7;
  30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15;
  ... (End)

Links

Crossrefs

Cf. A007931, A076478, A103897, A171757, A346303, A346304.

Programs

  • Mathematica
    (See A007931.)
A346306[rowmax_]:=Table[Range[2^(r+1)-2,2^r-1,-1],{r,rowmax}]; A346306[6] (* Paolo Xausa, Mar 09 2023 *)
  • Python
    from itertools import product
def comp(s): z, o = ord('0'), ord('1'); return s.translate({z:o, o:z})
def wgen(maxdigits):
    for digits in range(1, maxdigits+1):
        for b in product("01", repeat=digits):
            yield "".join(b)
def auptod(maxdigits):
    w = [None] + [wn for wn in wgen(maxdigits)]
    return [w.index(comp(w[n])) for n in range(1, 2**(maxdigits+1) - 1)]
print(auptod(6)) # Michael S. Branicky, Sep 03 2021

Formula

a(n) = 3*(2^d - 1) - n, where 2^d - 1 <= n <= 2^(d+1) - 2. - Michael S. Branicky, Sep 03 2021
Showing 1-3 of 3 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.