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.

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);

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).

A340666 A(n,k) is derived from n by replacing each 0 in its binary representation with a string of k 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 4, 3, 1, 0, 1, 8, 3, 4, 3, 0, 1, 16, 3, 16, 5, 3, 0, 1, 32, 3, 64, 9, 6, 7, 0, 1, 64, 3, 256, 17, 12, 7, 1, 0, 1, 128, 3, 1024, 33, 24, 7, 8, 3, 0, 1, 256, 3, 4096, 65, 48, 7, 64, 9, 3, 0, 1, 512, 3, 16384, 129, 96, 7, 512, 33, 10, 7
Offset: 0

Views

Author

Alois P. Heinz, Jan 15 2021

Keywords

Examples

			Square array A(n,k) begins:
  0, 0,  0,   0,    0,     0,      0,       0,        0, ...
  1, 1,  1,   1,    1,     1,      1,       1,        1, ...
  1, 2,  4,   8,   16,    32,     64,     128,      256, ...
  3, 3,  3,   3,    3,     3,      3,       3,        3, ...
  1, 4, 16,  64,  256,  1024,   4096,   16384,    65536, ...
  3, 5,  9,  17,   33,    65,    129,     257,      513, ...
  3, 6, 12,  24,   48,    96,    192,     384,      768, ...
  7, 7,  7,   7,    7,     7,      7,       7,        7, ...
  1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, ...
  ...
		

Crossrefs

Columns k=0-2, 4 give: A038573, A001477, A084471, A084473.
Rows n=0..17, 19 give: A000004, A000012, A000079, A010701, A000302, A000051(k+1), A007283, A010727, A001018, A087289, A007582(k+1), A062709(k+2), A164346, A181565(k+1), A005009, A181404(k+3), A001025, A199493, A253208(k+1).
Main diagonal gives A340667.

Programs

  • Maple
    A:= (n, k)-> Bits[Join](subs(0=[0$k][], Bits[Split](n))):
    seq(seq(A(n, d-n), n=0..d), d=0..12);
    # second Maple program:
    A:= proc(n, k) option remember; `if`(n<2, n,
         `if`(irem(n, 2, 'r')=1, A(r, k)*2+1, A(r, k)*2^k))
        end:
    seq(seq(A(n, d-n), n=0..d), d=0..12);
  • Mathematica
    A[n_, k_] := FromDigits[IntegerDigits[n, 2] /. 0 -> Sequence @@ Table[0, {k}], 2];
    Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 02 2021 *)

Formula

A000120(A(n,k)) = A000120(n) = log_2(A(n,0)+1).
A023416(A(n,k)) = k * A023416(n) for n >= 1.

A084474 Write n in binary and replace 0 with 0000.

Original entry on oeis.org

1, 10000, 11, 100000000, 100001, 110000, 111, 1000000000000, 1000000001, 1000010000, 1000011, 1100000000, 1100001, 1110000, 1111, 10000000000000000, 10000000000001, 10000000010000, 10000000011, 10000100000000
Offset: 1

Views

Author

Reinhard Zumkeller, May 27 2003

Keywords

Comments

a(n)=A007088(A084473(n)); in A084472: replace 0 with 00.

Programs

  • Mathematica
    Table[FromDigits[Flatten[IntegerDigits[n,2]/.{0->{0,0,0,0}}]],{n,20}] (* Harvey P. Dale, Feb 19 2013 *)
Showing 1-3 of 3 results.