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.

A143100 A007318 * A143098.

Original entry on oeis.org

1, 3, 4, 6, 13, 30, 64, 129, 256, 510, 1021, 2046, 4096, 8193, 16384, 32766, 65533, 131070, 262144, 524289, 1048576, 2097150, 4194301, 8388606, 16777216, 33554433, 67108864, 134217726, 268435453, 536870910, 1073741824, 2147483649, 4294967296, 8589934590
Offset: 1

Views

Author

Gary W. Adamson, Jul 24 2008

Keywords

Examples

			a(4) = 6 = (1, 3, 3, 1) dot (1, 2, -1, 2) = (1 + 6 - 3 + 2).

Links

Crossrefs

Cf. A143097, A143098, A143099.

Programs

  • Maple
    A143098 := proc(n) if(n=1)then return 1: elif(n mod 3 = 0)then return -1: else return 2: fi: end: A143100 := proc(n) return add(binomial(n-1,k-1)*A143098(k),k=1..n): end: seq(A143100(n),n=1..34); # Nathaniel Johnston, Apr 30 2011

Formula

Binomial transform of A143098: (1, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, ...).
From R. J. Mathar, Jul 31 2008: (Start)
G.f.: (3x^3 - 2x^2 - x + 1)*x/((x^2-x+1)*(2x-1)*(x-1)).
a(n) = -1 + 2^(n-1) + A057079(n-1). (End)

A143097 3*k - 2 interleaved with 3*k - 1 and 3*k.

Original entry on oeis.org

1, 2, 4, 3, 5, 7, 6, 8, 10, 9, 11, 13, 12, 14, 16, 15, 17, 19, 18, 20, 22, 21, 23, 25, 24, 26, 28, 27, 29, 31, 30, 32, 34, 33, 35, 37, 36, 38, 40, 39, 41, 43, 42, 44, 46, 45, 47, 49, 48, 50, 52, 51, 53, 55, 54, 56, 58, 57, 59, 61, 60, 62, 64, 63, 65, 67, 66
Offset: 1

Views

Author

Gary W. Adamson, Jul 24 2008

Keywords

Comments

First differences give A143098.
Binomial transform = A143099: (1, 3, 9, 22, 50, 113, 256, ...).

Examples

			Interleave 3 subsets:
  1,....4,.......7,......10,......13,......16,...
  ...2,.......5,.......8,......11,......14,...
  .........3,.......6,.......9,......12,...
  ...

Links

Crossrefs

Cf. A115302, A143098, A143099.
Cf. A083220 (n + (n mod 4)). - Zak Seidov, Feb 23 2017

Programs

  • Maple
    A143097 := proc(n) if(n<=1)then return n: elif(n mod 3 <= 1)then return n+1-2*(n mod 3): else return n: fi: end: seq(A143097(n), n=1..70); # Nathaniel Johnston, Apr 30 2011
  • Mathematica
    With[{nn=70},Join[{1},Riffle[Rest[Select[Range[nn],!Divisible[#,3]&]], Range[ 3,nn,3],3]]] (* Harvey P. Dale, May 06 2012 *)
Table[If[k == 1, 1, k - 1 + Mod[k - 1, 3]], {k, 100}] (* Zak Seidov, Feb 23 2017 *)

Formula

A permutation of the natural numbers: 3*k - 2 interleaved with 3*k - 1 and 3*k; k=1,2,3,...; given a(1) = 1. a(n) = n if the subset = 3*k - 1: (2, 5, 8, ...); a(n) = n+1 in 3*k - 2, k>1: (4, 7, 10, ...); and a(n) = (n-1) in 3*k: (3, 6, 9, ...).
G.f.: x(1+x+2x^2-2x^3+x^4)/((1-x)^2(1+x+x^2)). - R. J. Mathar, Sep 06 2008
a(n) = if(n==1, 1, (n-1) + (n-1) mod 3). - Zak Seidov, Feb 23 2017
For n>1, a(n) = n+2*sin(2*(n+1)*Pi/3)/sqrt(3). - Wesley Ivan Hurt, Sep 27 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 2 - 2*Pi/(3*sqrt(3)) - log(2)/3. - Amiram Eldar, Aug 21 2023

A143099 A007318 * A143097.

Original entry on oeis.org

1, 3, 9, 22, 50, 113, 256, 576, 1281, 2818, 6146, 13313, 28672, 61440, 131073, 278530, 589826, 1245185, 2621440, 5505024, 11534337, 24117250, 50331650, 104857601, 218103808, 452984832, 939524097, 1946157058, 4026531842, 8321499137, 17179869184, 35433480192
Offset: 1

Views

Author

Gary W. Adamson, Jul 24 2008

Keywords

Comments

A143100 = (1, 3, 4, 6, 13, 30, 64, 129, ...).

Examples

			a(4) = 22 = (1, 3, 3, 1) dot (1, 2, 4, 3) = (1 + 6 + 12 + 3).
a(4) = 22 = 2*a(3) + A143099(3) = 2*9 + 4, where 4 = A143100(3).

Links

Crossrefs

Cf. A143097, A143098, A143100.

Programs

  • Maple
    A143097 := proc(n) if(n<=1)then return n: elif(n mod 3 <= 1)then return n+1-2*(n mod 3): else return n: fi: end: A143099 := proc(n) return add(binomial(n-1,k-1)*A143097(k),k=1..n): end: seq(A143099(n),n=1..32); # Nathaniel Johnston, Apr 30 2011

Formula

Binomial transform of A143097: (1, 2, 4, 3, 5, 7, 6, 8, 10, 9, 11, ...). a(n) = 2*a(n-1) + A143100(n-1).
G.f.: x*(5*x^4-7*x^3+5*x^2-3*x+1)/((1-x)*(x^2-x+1)*(1-2*x)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 14 2009; [corrected by R. J. Mathar, Sep 16 2009]
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.