A164115 Expansion of (1 - x^5) / ((1 - x) * (1 - x^4)) in powers of x.

1, 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, 2

Offset: 0

Author

Michael Somos, Aug 10 2009

Keywords

• nonn
• mult
• easy

Comments

The sequence A107453 has the same terms but different offset.

Convolution inverse of A164116.

Decimal expansion of 11111/99990. - Elmo R. Oliveira, Feb 18 2024

Examples

			1 + x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + x^7 + 2*x^8 + x^9 + x^10 + ...
		

Links

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

Crossrefs

Cf. A107453, A138191, A164116, A164117.

Programs

Magma

m:=100; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x+x^2+x^3+x^4)/(1-x^4))); // G. C. Greubel, Sep 22 2018

Mathematica

CoefficientList[Series[(1+x+x^2+x^3+x^4)/(1-x^4), {x, 0, 100}], x] (* G. C. Greubel, Sep 22 2018 *)
LinearRecurrence[{0,0,0,1},{1,1,1,1,2},120] (* or *) PadRight[{1},120,{2,1,1,1}] (* Harvey P. Dale, Aug 24 2019 *)

PARI

{a(n) = 2 - (n==0) - (n%4>0)}

PARI

x='x+O('x^99); Vec((1-x^5)/((1-x)*(1-x^4))) \\ Altug Alkan, Sep 23 2018

Formula

Euler transform of length-5 sequence [ 1, 0, 0, 1, -1].

a(n) is multiplicative with a(2) = 1, a(2^e) = 2 if e>1, a(p^e) = 1 if p>2.

a(n) = (-1)^n * A164117(n).

a(4*n) = 2 unless n=0. a(2*n + 1) = a(4*n + 2) = 1.

a(-n) = a(n). a(n+4) = a(n) unless n=0 or n=-4.

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

a(n) = A138191(n+2), n>0. - R. J. Mathar, Aug 17 2009

Dirichlet g.f. (1+1/4^s)*zeta(s). - R. J. Mathar, Feb 24 2011

a(n) = (i^n + (-i)^n + (-1)^n + 5)/4 for n > 0 where i is the imaginary unit. - Bruno Berselli, Feb 25 2011