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 1-2 of 2 results.

A144864 a(n) = (4*16^(n-1)-1)/3.

Original entry on oeis.org

1, 21, 341, 5461, 87381, 1398101, 22369621, 357913941, 5726623061, 91625968981, 1466015503701, 23456248059221, 375299968947541, 6004799503160661, 96076792050570581, 1537228672809129301, 24595658764946068821, 393530540239137101141, 6296488643826193618261, 100743818301219097892181
Offset: 1

Views

Author

Artur Jasinski, Sep 23 2008

Keywords

Comments

Old name was: A144863, read as binary numbers, converted to base 10.
All numbers in this sequence for n>1 are congruent to 5 mod 16. - Artur Jasinski, Sep 25 2008
From Omar E. Pol, Sep 10 2011: (Start)
It appears that this is a bisection of A002450.
It appears that this is a bisection of A084241.
It appears that this is a bisection of A153497.
It appears that this is a bisection of A088556, if n>=2.
(End)
All of the above is trivially true. - Joerg Arndt, Aug 19 2014
The aerated sequence (b(n))n>=1 = [1, 0, 21, 0, 341, 0, 5461, 0, 87381, ...] is a fourth-order linear divisibility sequence; that is, a(n) divides a(m) whenever n divides m. It is the case P1 = 0, P2 = -9, Q = -4 of the 3-parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Aug 26 2022

Links

Crossrefs

Cf. A001025, A002450, A013776, A056830, A084241, A088556, A094028, A098704, A131865, A135576, A144863, A153497.
Third quadrisection of Jacobsthal numbers A001045; the other quadrisections are A195156 (first), A139792 (second), and A141060 (fourth).

Programs

  • Magma
    [16^n/12-1/3: n in [1..20]]; // Vincenzo Librandi, Aug 03 2011
  • Mathematica
    Table[1/3 (-1 + 16^(n - 1)) + 16^(n - 1), {n, 1, 17}] (* Artur Jasinski, Sep 25 2008 *)
LinearRecurrence[{17,-16},{1,21},20] (* Harvey P. Dale, Jun 29 2022 *)
  • PARI
    vector(66,n,(4*16^(n-1)-1)/3) \\ Joerg Arndt, Aug 19 2014

Formula

a(n) = 16^n/12 - 1/3; a(n) = 16*a(n-1) + 5, a(1)=1. - Artur Jasinski, Sep 25 2008
G.f.: x*(1+4*x) / ( (16*x-1)*(x-1) ). - R. J. Mathar, Jan 06 2011
a(n)=b such that Integral_{x=-Pi/2..Pi/2} (-1)^(n+1)*2^(2*n-3)*(cos((2*n-1)*x))/(5/4+sin(x)) dx = c+b*log(3). - Francesco Daddi, Aug 02 2011
a(n) = (2^(4*n-2)-1)/3. - Klaus Purath, Jan 31 2021
From Jianing Song, Aug 30 2022: (Start)
a(n) = A001045(4*n-2).
a(n+1) - a(n) = 10*A013776(n-1) = 20*A001025(n-1) for n >= 1.
a(n) = 10*A098704(n) + 1 = 20*A131865(n-2) + 1 for n >= 2. (End)
E.g.f.: (exp(16*x) - 4*exp(x) + 3)/12. - Stefano Spezia, Apr 18 2024

Extensions

New name from Joerg Arndt, Aug 19 2014

A084240 a(n) = -5*a(n-1) - 4*a(n-2), a(0)=1, a(1)=0.

Original entry on oeis.org

1, 0, -4, 20, -84, 340, -1364, 5460, -21844, 87380, -349524, 1398100, -5592404, 22369620, -89478484, 357913940, -1431655764, 5726623060, -22906492244, 91625968980, -366503875924, 1466015503700, -5864062014804, 23456248059220, -93824992236884, 375299968947540
Offset: 0

Views

Author

Paul Barry, May 23 2003

Keywords

Comments

A Jacobsthal related sequence.
Inverse binomial transform of A084246.

Links

Crossrefs

Cf. A001045, A002450, A080674, A084241.

Programs

  • Mathematica
    LinearRecurrence[{-5,-4},{1,0},30] (* Harvey P. Dale, Dec 13 2021 *)

Formula

G.f.: (1+5*x)/((1+x)*(1+4*x)).
E.g.f.: (4*exp(-x) - exp(-4*x))/3.
a(n) = A084241(n) + (-1)^n
a(n) = (-1)^(n+1)*(A002450(n)-1) = (-1)^(n+1)*(A001045(2*n)-1).
a(n) = (4*(-1)^n - (-4)^n)/3.
a(n) = (-1)^(n-1) * A080674(n-1). - Falk Hüffner, Jan 11 2021
Showing 1-2 of 2 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.