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

0,5

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

Links

Table of n, a(n) for n=0..104.

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

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A107453, A138191, A164116, A164117.

Sequence in context: A186006 A236398 A107453 * A164117 A177704 A138191

Adjacent sequences: A164112 A164113 A164114 * A164116 A164117 A164118

Keyword

nonn,mult,easy

Author

Michael Somos, Aug 10 2009

Status

approved