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

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

Offset: 1

Author

Artur Jasinski, Sep 23 2008

Keywords

• easy
• nonn

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

• Vincenzo Librandi, Table of n, a(n) for n = 1..500
• H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
• Index entries for linear recurrences with constant coefficients, signature (17,-16).

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