A163817 -id:A163817 - cp's OEIS frontend

A163811 Expansion of (1 - x) * (1 - x^10) / ((1 - x^5) * (1 - x^6)) in powers of x.

1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0

Offset: 0

Michael Somos, Aug 04 2009, Aug 09 2009

			1 - x + x^5 - x^7 + x^11 - x^13 + x^17 - x^19 + x^23 - x^25 + x^29 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, -1, 0, -1).

A163817(n) = -a(n) unless n=0. A163817(n) = (-1)^n * a(n).

Convolution inverse of A163812.

Join[{1}, LinearRecurrence[{0, -1, 0, -1}, {-1, 0, 0, 0}, 50]] (* G. C. Greubel, Aug 04 2017 *)

{a(n) = (n==0) + [0, -1, 0, 0, 0, 1][n%6 + 1]}

PARI
```
{a(n) = (n==0) - kronecker(-12, n)}
```

Euler transform of length 10 sequence [ -1, 0, 0, 0, 1, 1, 0, 0, 0, -1].
a(2*n) = a(3*n) = 0 unless n=0, a(6*n + 1) = -1, a(6*n + 5) = a(0) = 1.
a(-n) = -a(n) unless n=0. a(n+6) = a(n) unless n=0 or n=-6.
G.f.: (1 - x + x^2 - x^3 + x^4) / (1 + x^2 + x^4).
G.f. A(x) = 1 - x / ( 1 + x^4 / (1 + x^2)) = 1 / (1 + x / (1 - x / (1 + x^3 / (1 + x^2 / (1 + x / (1 - x)))))). - Michael Somos, Jan 03 2013

A163818 Expansion of (1 - x) * (1 - x^6) / ((1 - x^2) * (1 - x^5)) in powers of x.

Original entry on oeis.org

1, -1, 1, -1, 1, 0, -1, 1, -1, 1, 0, -1, 1, -1, 1, 0, -1, 1, -1, 1, 0, -1, 1, -1, 1, 0, -1, 1, -1, 1, 0, -1, 1, -1, 1, 0, -1, 1, -1, 1, 0, -1, 1, -1, 1, 0, -1, 1, -1, 1, 0, -1, 1, -1, 1, 0, -1, 1, -1, 1, 0, -1, 1, -1, 1, 0, -1, 1, -1, 1, 0, -1, 1, -1, 1, 0, -1, 1, -1, 1, 0, -1, 1, -1, 1, 0, -1, 1, -1, 1, 0, -1, 1, -1, 1, 0, -1, 1, -1, 1, 0, -1, 1, -1, 1

Offset: 0

Michael Somos, Aug 04 2009

			G.f. = 1 - x + x^2 - x^3 + x^4 - x^6 + x^7 - x^8 + x^9 - x^11 + x^12 - x^13 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1, -1, -1, -1).

Cf. A163813, A163817.

a[ n_] := Boole[n == 0] + {-1, 1, -1, 1, 0}[[Mod[n, 5, 1]]]; (* Michael Somos, Jun 17 2015 *)
a[ n_] := Boole[n == 0] + (-1)^(n + Quotient[n, 5]) Sign@Mod[n, 5]; (* Michael Somos, Jun 17 2015 *)
CoefficientList[Series[(1 + x^2 + x^4) / (1 + x + x^2 + x^3 + x^4), {x, 0, 100}], x] (* Vincenzo Librandi, Aug 05 2017 *)

{a(n) = (n==0) + [ 0, -1, 1, -1, 1][n%5 + 1]};

{a(n) = (n==0) + (-1)^(n + n\5) * kronecker(25, n)};

Euler transform of length 6 sequence [-1, 1, 0, 0, 1, -1].
a(5*n) = 0 unless n=0. a(5*n + 1) = a(5*n + 3) = -1, a(5*n + 2) = a(5*n + 4) = a(0) = 1.
a(n) = -a(-n) unless n=0. a(n+5) = a(n) unless n=0 or n=-5.
G.f.: (1 + x^2 + x^4) / (1 + x + x^2 + x^3 + x^4).
G.f.: 1 / (1 + x / (1 + x^4 / (1 + x^2))). - Michael Somos, Jan 03 2013
a(n) = (-1)^n * A163812(n). Convolution inverse of A163817.

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A163811 Expansion of (1 - x) * (1 - x^10) / ((1 - x^5) * (1 - x^6)) in powers of x.

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A163818 Expansion of (1 - x) * (1 - x^6) / ((1 - x^2) * (1 - x^5)) in powers of x.

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