A100285 Expansion of (1+5*x^2)/(1-x+x^2-x^3).

1, 1, 5, 5, 1, 1, 5, 5, 1, 1, 5, 5, 1, 1, 5, 5, 1, 1, 5, 5, 1, 1, 5, 5, 1, 1, 5, 5, 1, 1, 5, 5, 1, 1, 5, 5, 1, 1, 5, 5, 1, 1, 5, 5, 1, 1, 5, 5, 1, 1, 5, 5, 1, 1, 5, 5, 1, 1, 5, 5, 1, 1, 5, 5, 1, 1, 5, 5, 1, 1, 5, 5, 1, 1, 5, 5, 1, 1, 5, 5, 1, 1, 5, 5, 1, 1, 5, 5, 1, 1

Offset: 0

Author

Paul Barry, Nov 11 2004

Keywords

• nonn
• easy

Comments

This sequence is periodic. - T. D. Noe, Nov 09 2006

Decimal expansion of 35/303. - Elmo R. Oliveira, May 11 2024

Links

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

Crossrefs

Cf. A056594, A100284, A133872.

Programs

Magma

[(((n+2) mod 4) + 5*(n mod 4) - 6*(n mod 2))/2: n in [0..100]]; // G. C. Greubel, Feb 06 2023

Mathematica

CoefficientList[Series[(1+5x^2)/(1-x+x^2-x^3),{x,0,100}],x] (* or *) PadRight[{},100,{1,1,5,5}] (* Harvey P. Dale, Jun 02 2021 *)

SageMath

def A100285(n): return (((n+2)%4) +5*(n%4) -6*(n%2))/2
[A100285(n) for n in range(101)] # G. C. Greubel, Feb 06 2023

Formula

a(n) = a(n-1) - a(n-2) + a(n-3)

a(n) = 3 - 2*cos(Pi*n/2) - 2*sin(Pi*n/2).

a(n) = mod(A100284(n), 8).

From G. C. Greubel, Feb 06 2023: (Start)

a(n) = A133872(n) + 5*A133872(n+2).

a(n) = ((n+2 mod 4) + 5*(n mod 4) - 6*(n mod 2))/2.

a(n) = 3 -((1+i)*(-1)^n +(1-i)*i^n) = 3 -2*(A056594(n) +A056594(n-1)).

G.f.: (1+5*x^2)/((1-x)*(1+x^2)).

E.g.f.: 3*exp(x) - 2*cos(x) - 2*sin(x). (End)

Extensions

Corrected by T. D. Noe, Nov 09 2006