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.

A107687 Complement of A107686.

Original entry on oeis.org

6, 12, 13, 14, 22, 24, 25, 28, 29, 30, 38, 44, 45, 46, 48, 49, 54, 56, 57, 60, 61, 62, 70, 76, 77, 78, 86, 88, 89, 92, 93, 94, 96, 97, 102, 108, 109, 110, 112, 113, 118, 120, 121, 124, 125, 126, 134, 140, 141, 142, 150, 152, 153, 156, 157, 158, 166, 172, 173, 174, 176
Offset: 1

Views

Author

Reinhard Zumkeller, May 20 2005

Keywords

Extensions

Corrected by T. D. Noe, Oct 25 2006

A107688 A107686(n+1) - A107686(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 2, 3, 1, 4, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 3, 1, 1, 1, 2, 3, 1, 4, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 2, 3, 1, 4, 3, 1, 1, 1, 2, 1, 1, 1, 1, 4, 3, 1, 1, 1, 2, 3, 1, 4, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 2, 3, 1, 4, 1, 1
Offset: 0

Views

Author

Reinhard Zumkeller, May 20 2005

Keywords

Comments

a(n) <= 4.

A107750 If n=0 then 0, else smallest number greater than its predecessor and having either more or fewer zeros in its binary representation.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 91, 92, 93
Offset: 0

Views

Author

Reinhard Zumkeller, May 23 2005

Keywords

Comments

Essentially the complement of A016825 with respect to the nonnegative integers (except for 2). A023416(a(n+1)) <> A023416(a(n)).

Links

Crossrefs

Cf. A107686, A007088, A023416.

Programs

  • Haskell
    a107750 n = a107750_list !! n
a107750_list = 0 : f 0 where
   f x = y : f y where
     y = head [z | z <- [x + 1 ..], a023416 z /= a023416 x]
-- Reinhard Zumkeller, Jul 07 2014
  • Mathematica
    Table[n - Sign[Floor[n/3]] + Floor[(1/2) Sum[Ceiling[(i + 2)/3] - Floor[(i + 2)/3], {i, n}]], {n, 0, 50}] (* Wesley Ivan Hurt, Jun 16 2014 *)

Formula

a(n+1) = a(n) + A107751(n).
For k >= 0, 0 <= i <= 3*2^k:
a(6*2^k + i) = a(3*2^k + i) + 4*2^k,
a(9*2^k + i) = a(3*2^k + i) + 8*2^k.
a(n) = n - sign(floor(n/3)) + floor( (1/2)*sum_{i=1..n} ( ceiling((i+2)/3) - floor((i+2)/3) ) ). - Wesley Ivan Hurt, Jun 16 2014
Conjectures from Colin Barker, Jul 24 2017: (Start)
G.f.: x*(1+x)*(1+x^2-x^3+x^4) / ((1-x)^2*(1+x+x^2)).
a(n) = a(n-1) + a(n-3) - a(n-4) for n>3.
(End)

A107684 Union of sequences 2^k-1, 2^k and 2^k+1.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 15, 16, 17, 31, 32, 33, 63, 64, 65, 127, 128, 129, 255, 256, 257, 511, 512, 513, 1023, 1024, 1025, 2047, 2048, 2049, 4095, 4096, 4097, 8191, 8192, 8193, 16383, 16384, 16385, 32767, 32768, 32769, 65535, 65536, 65537, 131071
Offset: 0

Views

Author

Reinhard Zumkeller, May 20 2005

Keywords

Comments

Subsequence of A107686.

Links

Crossrefs

Cf. A000225, A000079, A000051.

Programs

  • Mathematica
    Flatten[{#-1,#,#+1}&/@(2^Range[0,20])]//Union (* Harvey P. Dale, Oct 06 2017 *)

Formula

a(n) = if n<=2 then n else 2^(floor(n/3)+1) + n mod 3 - 1. - Reinhard Zumkeller, Jun 05 2005
G.f.:-x*(1+3*x+6*x^2+7*x^3+6*x^4+4*x^5+2*x^6)/((2*x^3-1)*(x^2+x+1)) [From Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009]

Extensions

Description corrected by Henrik Lundquist, Jun 06 2005
Showing 1-4 of 4 results.

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