A138191 Denominator of (n-1)*n*(n+1)/12.

1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Author

Eric W. Weisstein, Mar 04 2008

Keywords

• nonn
• frac
• mult
• easy

Comments

Proof of 4-periodicity follows from evaluating (n+3)(n+4)(n+5)/12, subtracting (n-1)n(n+1)/12 and getting n^2+4n+5 which is an integer. - R. J. Mathar, Mar 07 2008

Examples

			0, 1/2, 2, 5, 10, 35/2, 28, 42, 60, 165/2, 110, 143, 182, ...
		

Links

• Eric Weisstein's World of Mathematics, Kirchhoff Index.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Crossrefs

Cf. A000292, A107453, A107453, A138190.

Programs

Mathematica

Table[(n^3-n)/12,{n,120}]//Denominator (* or *) PadRight[{},120,{1,2,1,1}] (* Harvey P. Dale, Apr 15 2019 *)

Python

def A138191(n): return (1,1,2,1)[n&3] # Chai Wah Wu, Apr 25 2024

Formula

From R. J. Mathar, Mar 07 2008: (Start)

a(n) = 1 + (A000292(n-1) mod 2) = a(n-4).

O.g.f.: -1-5/(4(x-1))+1/(4(x+1))-1/(2(x^2+1)). (End)

From Amiram Eldar, Jan 01 2023: (Start)

Multiplicative with a(p^e) = 2 if p = 2 and e = 1, and 1 otherwise.

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

Sum_{k=1..n} a(k) ~ (5/4)*n. (End)