A112132 Period 4: repeat [1, 3, 1, 7].

1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1, 7, 1, 3, 1

Offset

1,2

Links

Table of n, a(n) for n=1..87.

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

Formula

a(n+1) = 3-2*sin(Pi*n/2)-2*(-1)^n. - R. J. Mathar, Oct 08 2011

Multiplicative with a(2) = 3, a(2^e) = 7 if e >= 2, a(p^e) = 1 otherwise. - Antti Karttunen, Mar 31 2013, typo corrected May 02 2020

From Wesley Ivan Hurt, Jul 09 2016: (Start)

G.f.: x*(1+3*x+x^2+7*x^3)/(1-x^4).

a(n) = a(n-4) for n>4.

a(2n) = 5+2*(-1)^n, a(2n-1) = 1. (End)

Dirichlet g.f.: zeta(s)*(1+2^(1-s)+4^(1-s)). - Amiram Eldar, Jan 03 2023

From Ridouane Oudra, Sep 14 2025: (Start)

a(n) = Sum_{d|n} tau(d)*A085384(n/d).

a(n) = 2*gcd(n,4) - 1.

a(n) = sigma(gcd(n,4)). (End)

E.g.f.: 2*cos(x) + 5*cosh(x) + sinh(x) - 7. - Amiram Eldar, Dec 15 2025

Maple

seq(op([1, 3, 1, 7]), n=1..50); # Wesley Ivan Hurt, Jul 09 2016

Mathematica

PadRight[{}, 100, {1, 3, 1, 7}] (* Wesley Ivan Hurt, Jul 09 2016 *)

Prog

(PARI) a(n)=1+2*((n-1)%2)*((n-1)%4); \\ Jaume Oliver Lafont, Aug 28 2009; corrected by Antti Karttunen, Mar 31 2013
(PARI) a(n)=[1, 3, 1, 7][1+(n-1)%4]; \\ Joerg Arndt, Apr 02 2013
(PARI) A112132(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i, 1], if(1==f[i, 2], 3, 7), 1)); }; \\ (implements the multiplicative formula) - Antti Karttunen, May 10 2020
(Magma) &cat [[1, 3, 1, 7]^^30]; // Wesley Ivan Hurt, Jul 09 2016

Crossrefs

First differences of A112062.

Also half of the first differences of A112072. Cf. A112086.

Cf. A109008, A000005, A000203, A085384.

Sequence in context: A136011 A227984 A021991 * A053381 A038712 A354587

Adjacent sequences: A112129 A112130 A112131 * A112133 A112134 A112135

Keyword

nonn,mult,easy

Author

Antti Karttunen, Aug 28 2005

Status

approved