A146160 Period 4: repeat [1, 4, 1, 16].

1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1

Offset: 1

Author

Artur Jasinski, Oct 27 2008

Keywords

• nonn
• easy
• mult

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Crossrefs

Cf. A010156, A145996. [Artur Jasinski, Oct 29 2008]

Programs

Magma

&cat[[1,4,1,16]^^20]; // Vincenzo Librandi, Feb 04 2016

Maple

A146160:=n->[1, 4, 1, 16][(n mod 4)+1]: seq(A146160(n), n=0..100); # Wesley Ivan Hurt, Jun 15 2016

Mathematica

Table[GCD[4k - k^2, 5k^2, 20k - 20k^2, 16 - 32k + 16k^2], {k, 100}]
PadRight[{},100,{1,4,1,16}] (* or *) LinearRecurrence[{0,0,0,1},{1,4,1,16},90] (* Harvey P. Dale, Mar 29 2025 *)

PARI

Vec((1+4*x+x^2+16*x^3)/(1-x^4) + O(x^100)) \\ Altug Alkan, Feb 04 2016

Formula

Continued fraction of (8 + sqrt(78))/14.

GCD(4k - k^2, 5k^2, 20k - 20k^2, 16 - 32k + 16k^2) for k = 1,2,3,...

From Artur Jasinski, Oct 29 2008: (Start)

a(n) = 1 when n congruent to 1 or 3 mod 4.

a(n) = 4 when n congruent to 2 mod 4.

a(n) = 16 when n congruent to 0 mod 4. (End)

From Richard Choulet, Nov 03 2008: (Start)

a(n+4) = a(n).

a(n) = (9/2)*(-1)^n + (11/2) + 6*cos(Pi*n/2).

O.g.f.: f(z) = a(0)+a(1)*z+... = (1+4*z+z^2+16*z^3)/(1-z^4). (End)

E.g.f.: sinh(x) + 20*(sinh(x/2))^2 - 12*(sin(x/2))^2. - G. C. Greubel, Feb 03 2016

a(n) = a(-n). - Wesley Ivan Hurt, Jun 15 2016

a(n) = A109008(n)^2. - R. J. Mathar, Feb 12 2019

From Amiram Eldar, Jan 01 2023: (Start)

Multiplicative with a(2) = 4, a(2^e) = 16 for e >= 2, and a(p^e) = 1 for p >= 3.

Dirichlet g.f.: zeta(s)*(12/4^s+3/2^s+1). (End)

Extensions

Choulet formula adapted for offset 1 from Wesley Ivan Hurt, Jun 15 2016