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 1-4 of 4 results.

A038573 a(n) = 2^A000120(n) - 1.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 3, 7, 1, 3, 3, 7, 3, 7, 7, 15, 1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31, 1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31, 3, 7, 7, 15, 7, 15, 15, 31, 7, 15, 15, 31, 15, 31, 31, 63, 1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31, 3, 7, 7, 15, 7, 15, 15, 31
Offset: 0

Views

Author

Marc LeBrun

Keywords

Comments

Essentially the same sequence as A001316, which has much more information, and also A159913. - N. J. A. Sloane, Jun 05 2009
Smallest number with same number of 1's in its binary expansion as n.
Fixed point of the morphism 0 -> 01, 1 -> 13, 3 -> 37, ... = k -> k, 2k+1, ... starting from a(0) = 0; 1 -> 01 -> 0113 -> 01131337 -> 011313371337377(15) -> ..., . - Robert G. Wilson v, Jan 24 2006
From Gary W. Adamson, Jun 04 2009: (Start)
As an infinite string, 2^n terms per row starting with "1": (1; 1,3; 1,3,3,7; 1,3,3,7,3,7,7,15; 1,3,3,7,3,7,7,15,3,7,7,15,7,15,15,31;...)
Row sums of that triangle = A027649: (1, 4, 14, 46, 454, ...); where the next row sum = current term of A027649 + next term in finite difference row of A027649, i.e., (1, 3, 10, 32, 100, 308, ...) = A053581. (End)
From Omar E. Pol, Jan 24 2016: (Start)
Partial sums give A267700.
a(n) is also the number of cells turned ON at n-th generation of the cellular automaton of A267700 in a 90-degree sector on the square grid.
a(n) is also the number of Y-toothpicks added at n-th generation of the structure of A267700 in a 120-degree sector on the triangular grid. (End)
Row sums of A090971. - Nikolaos Pantelidis, Nov 23 2022

Examples

			9 = 1001 -> 0011 -> 3, so a(9)=3.
From _Gary W. Adamson_, Jun 04 2009: (Start)
Triangle read by rows:
  1;
  1, 3;
  1, 3, 3, 7;
  1, 3, 3, 7, 3, 7, 7, 15;
  1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31;
  ...
Row sums: (1, 4, 14, 46, ...) = A027649 = last row terms + new set of terms such that row 3 = (1, 3, 3, 7,) + (3, 7, 7, 15) = 14 + 32 = A027649(2) + A053581(3). (End)
The rows of this triangle converge to A159913. - _N. J. A. Sloane_, Jun 05 2009
G.f. = x + x^2 + 3*x^3 + x^4 + 3*x^5 + 3*x^6 + 7*x^7 + x^8 + 3*x^9 + 3*x^10 + 7*x^11 + ... - _Michael Somos_, Jul 24 2023

Links

Crossrefs

Cf. A007313, A115378, A073138.
This is Guy Steele's sequence GS(3, 6) (see A135416).
Cf. also A000079, A001316, A027649, A053581, A159913, A267700.
Write n in b-ary, sort digits into increasing order: this sequence (b=2), A038574 (b=3), A319652 (b=4), A319653 (b=5), A319654 (b=6), A319655 (b=7), A319656 (b=8), A319657 (b=9), A004185 (b=10).
Column k=0 of A340666.

Programs

  • Haskell
    a038573 0 = 0
a038573 n = (m + 1) * (a038573 n') + m where (n', m) = divMod n 2
-- Reinhard Zumkeller, Oct 10 2012, Feb 07 2011
(Python 3.10+)
def A038573(n): return (1<Chai Wah Wu, Nov 15 2022
  • Maple
    seq(2^convert(convert(n,base,2),`+`)-1, n=0..100); # Robert Israel, Jan 24 2016
  • Mathematica
    Array[ 2^Count[ IntegerDigits[ #, 2 ], 1 ]-1&, 100 ]
Nest[ Flatten[ # /. a_Integer -> {a, 2a + 1}] &, {0}, 7] (* Robert G. Wilson v, Jan 24 2006 *)
  • PARI
    {a(n) = 2^subst(Pol(binary(n)), x, 1) - 1};
  • PARI
    a(n) = 2^hammingweight(n)-1; \\ Michel Marcus, Jan 24 2016

Formula

a(2n) = a(n), a(2n+1) = 2*a(n)+1, a(0) = 0. a(n) = A001316(n)-1 = 2^A000120(n) - 1. - Daniele Parisse
a(n) = number of positive integers k < n such that n XOR k = n-k (cf. A115378). - Paul D. Hanna, Jan 21 2006
a(n) = f(n, 1) with f(x, y) = if x = 0 then y - 1 else f(floor(x/2), y*(1 + x mod 2)). - Reinhard Zumkeller, Nov 21 2009
a(n) = (n mod 2 + 1) * a(floor(n/2)) + n mod 2. - Reinhard Zumkeller, Oct 10 2012
a(n) = Sum_{i=1..n} C(n,i) mod 2. - Wesley Ivan Hurt, Nov 17 2017
G.f.: -1/(1 - x) + Product_{k>=0} (1 + 2*x^(2^k)). - Ilya Gutkovskiy, Aug 20 2019
G.f. A(x) = x + x^2*A(x) + (1 + 2*x)*(1 - x^2)*A(x^2). - Michael Somos, Jul 24 2023

Extensions

More terms from Erich Friedman
New definition from N. J. A. Sloane, Mar 01 2008

A084471 Change 0 to 00 in binary representation of n.

Original entry on oeis.org

1, 4, 3, 16, 9, 12, 7, 64, 33, 36, 19, 48, 25, 28, 15, 256, 129, 132, 67, 144, 73, 76, 39, 192, 97, 100, 51, 112, 57, 60, 31, 1024, 513, 516, 259, 528, 265, 268, 135, 576, 289, 292, 147, 304, 153, 156, 79, 768, 385, 388, 195, 400, 201, 204, 103, 448, 225
Offset: 1

Views

Author

Reinhard Zumkeller, May 27 2003

Keywords

Comments

a(n) = n iff n = 2^k - 1, k>0.
A023416(a(n))=A023416(n)*2; A000120(a(n))=A000120(n);

Links

Crossrefs

Cf. A084472(n)=A007088(a(n)), A084473, A038573.
Ordered terms are in A060142.
Column k=2 of A340666.
Cf. A088698, A175047. - Robert G. Wilson v, Dec 10 2009

Programs

  • Haskell
    a084471 1 = 1
a084471 x = 2 * (2 - d) * a084471 x' + d  where (x',d) = divMod x 2
-- Reinhard Zumkeller, Jul 16 2012
  • Maple
    a:= n-> Bits[Join](subs(0=[0$2][], Bits[Split](n))):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 15 2021
  • Mathematica
    f[n_] := FromDigits[Flatten[IntegerDigits[n, 2] /. {0 -> {0, 0}}], 2]; Array[f, 60] (* Robert G. Wilson v, Dec 10 2009 *)

Formula

a(1)=1, a(2*k+1)=2*a(k)+1, a(2*k)=4*a(k).

A084473 Replace 0 with 0000 in binary representation of n.

Original entry on oeis.org

1, 16, 3, 256, 33, 48, 7, 4096, 513, 528, 67, 768, 97, 112, 15, 65536, 8193, 8208, 1027, 8448, 1057, 1072, 135, 12288, 1537, 1552, 195, 1792, 225, 240, 31, 1048576, 131073, 131088, 16387, 131328, 16417, 16432, 2055, 135168, 16897, 16912, 2115, 17152, 2145
Offset: 1

Views

Author

Reinhard Zumkeller, May 27 2003

Keywords

Comments

a(n) = n iff n = 2^k - 1, k>0 (A000225). - Bernard Schott, Dec 18 2021

Links

Crossrefs

Cf. A000120, A000225, A084471, A084474, A007088, A023416.
Column k=4 of A340666.

Programs

  • Haskell
    a084473 1 = 1
a084473 x = 2 * (if b == 1 then 1 else 8) * a084473 x' + b
            where (x', b) = divMod x 2
-- Reinhard Zumkeller, Mar 31 2015
  • Maple
    a:= n-> Bits[Join](subs(0=[0$4][], Bits[Split](n))):
seq(a(n), n=1..49);  # Alois P. Heinz, Jan 15 2021
  • Mathematica
    a[n_] := FromDigits[IntegerDigits[n, 2] /. 0 -> Sequence@@{0,0,0,0}, 2];
Array[a, 50] (* Jean-François Alcover, Dec 16 2021 *)
  • Python
    def a(n): return int(bin(n)[2:].replace('0', '0000'), 2)
print([a(n) for n in range(1, 46)]) # Michael S. Branicky, Jan 15 2021

Formula

a(1)=1, a(2*k+1)=2*a(k)+1, a(2*k)=16*a(k).
a(n) = A084471(A084471(n)).
A084474(n) = A007088(a(n));
A023416(a(n)) = A023416(n)*4.
A000120(a(n)) = A000120(n).

A340667 a(n) is derived from n by replacing each 0 in its binary representation with a string of n 0's.

Original entry on oeis.org

0, 1, 4, 3, 256, 65, 192, 7, 16777216, 524289, 2098176, 8195, 50331648, 49153, 114688, 15, 18446744073709551616, 4503599627370497, 36028797019226112, 1099511627779, 2305844108725321728, 17592190238721, 70368756760576, 67108871, 14167099448608935641088
Offset: 0

Views

Author

Alois P. Heinz, Jan 15 2021

Keywords

Links

Crossrefs

Main diagonal of A340666.
Cf. A000079, A000120, A000225, A023416, A057156.

Programs

  • Maple
    a:= n-> Bits[Join](subs(0=[0$n][], Bits[Split](n))):
seq(a(n), n=0..25);
# second Maple program:
b:= proc(n, k) option remember; `if`(n<2, n,
     `if`(irem(n, 2, 'r')=1, b(r, k)*2+1, b(r, k)*2^k))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);
  • Python
    def A340667(n):
    return 0 if n == 0 else int(bin(n)[2:].replace('0','0'*n),2) # Chai Wah Wu, Jan 29 2021

Formula

a(n) = n <=> n in { A000225 }.
a(n) = n^n <=> n in { A000079 }.
A000120(a(n)) = A000120(n).
A023416(a(n)) = n * A023416(n) for n >= 1.
Showing 1-4 of 4 results.

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

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