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.

A140659 a(n) = floor(A140657(n+2)/10).

Original entry on oeis.org

0, 0, 1, 2, 6, 12, 25, 50, 102, 204, 409, 818, 1638, 3276, 6553, 13106, 26214, 52428, 104857, 209714, 419430, 838860, 1677721, 3355442, 6710886, 13421772, 26843545, 53687090, 107374182, 214748364, 429496729, 858993458, 1717986918, 3435973836
Offset: 0

Views

Author

Paul Curtz, Jul 10 2008

Keywords

Links

Crossrefs

Cf. A093387.

Programs

  • Mathematica
    a[ n_] := If[n < 2, 0, 2 a[n - 1] + If[ EvenQ[n], Mod[n/2, 2, 1], 0]]; (* Michael Somos, Mar 02 2014 *)
  • PARI
    f(n) = 2^n+3*(-1)^n; \\ A140657
a(n) = f(n+2)\10; \\ Michel Marcus, Sep 08 2019

Formula

a(2n+2) = 4*a(2n)+A000034(n).
a(2n+1) = 2*a(2n).

Extensions

Edited and extended by R. J. Mathar, Jul 29 2008
Name corrected by Michel Marcus, Sep 08 2019

A140683 a(n) = 3*(-1)^(n+1)*2^n - 1.

Original entry on oeis.org

-4, 5, -13, 23, -49, 95, -193, 383, -769, 1535, -3073, 6143, -12289, 24575, -49153, 98303, -196609, 393215, -786433, 1572863, -3145729, 6291455, -12582913, 25165823, -50331649, 100663295, -201326593, 402653183, -805306369, 1610612735, -3221225473
Offset: 0

Views

Author

Paul Curtz, Jul 11 2008

Keywords

Comments

Alternated reading of negative of A140660 and A140529.
The binomial transform yields -4 followed by the negative of A140657.
The inverse binomial transform yields essentially a signed version of A000244. - R. J. Mathar, Aug 02 2008

Links

Programs

  • Magma
    [3*(-1)^(n+1)*2^n-1: n in [0..40]]; // Vincenzo Librandi, Aug 08 2011
  • Mathematica
    Table[3(-1)^(n+1)2^n-1,{n,0,40}] (* or *) LinearRecurrence[{-1,2},{-4,5},40] (* Harvey P. Dale, May 26 2011 *)

Formula

a(2n) = -A140660(n). a(2n+1) = A140529(n).
a(n+1) - a(n) = (-1)^n*A005010(n). a(2n) + a(2n+1) = A096045(n).
a(n) = A140590(n+1) - 2*A140590(n).
O.g.f: (4-x)/((x-1)(2x+1)). - R. J. Mathar, Aug 02 2008
a(n) = -a(n-1) + 2*a(n-2); a(0)=-4, a(1)=5. - Harvey P. Dale, May 26 2011

Extensions

Edited and extended by R. J. Mathar, Aug 02 2008

A280345 a(0) = 3, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [1, -2].

Original entry on oeis.org

3, 7, 12, 25, 48, 97, 192, 385, 768, 1537, 3072, 6145, 12288, 24577, 49152, 98305, 196608, 393217, 786432, 1572865, 3145728, 6291457, 12582912, 25165825, 50331648, 100663297, 201326592, 402653185, 805306368, 1610612737, 3221225472, 6442450945, 12884901888
Offset: 0

Views

Author

Paul Curtz, Jan 01 2017

Keywords

Comments

a(n) mod 9 is a periodic sequence of length 2: repeat [3, 7].
From 7, the last digit is of period 4: repeat [7, 2, 5, 8].
(Main sequence for the signature (2,1,-2): 0, 0, 1, 2, 5, 10, 21, 42, ... = 0 followed by A000975(n) = b(n), which first differences are A001045(n) (Paul Barry, Oct 08 2005). Then, 0 followed by b(n) is an autosequence of the first kind. The corresponding autosequence of the second kind is 0, 0, 2, 3, 8, 15, 32, 63, ... . See A277078(n).)
Difference table of a(n):
3, 7, 12, 25, 48, 97, 192, ...
4, 5, 13, 23, 49, 95, 193, ... = -(-1)^n* A140683(n)
1, 8, 10, 26, 46, 98, 190, ... = A259713(n)
7, 2, 16, 20, 52, 92, 196, ...
-5, 14, 4, 32, 40, 104, 184, ...
... .

Examples

			a(0) = 3, a(1) = 2*3 + 1 = 7, a(2) = 2*7 - 2 = 12, a(3) = 2*12 + 1 = 25.

Links

Crossrefs

Cf. A005010, A051049, A140657, A140683, A164346, A199116, A259713.

Programs

  • Mathematica
    a[0] = 3; a[n_] := a[n] = 2 a[n - 1] + 1 + (-3) Boole[EvenQ@ n]; Table[a@ n, {n, 0, 32}] (* or *)
CoefficientList[Series[(3 + x - 5 x^2)/((1 - x) (1 + x) (1 - 2 x)), {x, 0, 32}], x] (* Michael De Vlieger, Jan 01 2017 *)
  • PARI
    Vec((3 + x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Jan 01 2017

Formula

a(2n) = 3*4^n, a(2n+1) = 6*4^n + 1.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), n>2.
a(n+2) = a(n) + 9*2^n.
a(n) = 2^(n+2) - A051049(n).
From Colin Barker, Jan 01 2017: (Start)
a(n) = 3*2^n for n even.
a(n) = 3*2^n + 1 for n odd.
G.f.: (3 + x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)).
(End)
Binomial transform of 3, followed by (-1)^n* A140657(n).

Extensions

More terms from Colin Barker, Jan 01 2017
Showing 1-3 of 3 results.

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