A163811 -id:A163811 - cp's OEIS frontend

A163812 Expansion of (1 - x^5) * (1 - x^6) / ((1 - x) * (1 - x^10)) in powers of x.

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. A099443, A163811, A163818.

a[ n_] := Boole[n == 0] + (-1)^Quotient[n, 5] Sign@Mod[n, 5]; (* Michael Somos, Jun 17 2015 *)

{a(n) = (n==0) + [0, 1, 1, 1, 1, 0, -1, -1, -1, -1][n%10 + 1]};

Euler transform of length 10 sequence [ 1, 0, 0, 0, -1, -1, 0, 0, 0, 1].
a(5*n) = 0 unless n=0.
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).
a(n) = (-1)^n * A163818(n). Convolution inverse of A163811.
G.f.: A(x) = 1 / (1 - x / ( 1 + x^4 / (1 + x^2))) = 1 + x / (1 - x / (1 + x^3 / (1 + x^2 / (1 + x / (1 - x))))). - Michael Somos, Jan 03 2013
a(n) = A099443(n-1), n>0. - R. J. Mathar, Aug 05 2009

A163817 Expansion of (1 - x^2) * (1 - x^5) / ((1 - x) * (1 - x^6)) in powers of x.

Original entry on oeis.org

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

			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).

A163811(n) = -a(n) unless n=0. A163811(n) = (-1)^n * a(n).

Convolution inverse of A163818.

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 6 sequence [ 1, -1, 0, 0, -1, 1].
a(n) is multiplicative with a(2^e) = a(3^e) = 0^e, a(p^e) = 1 if p == 1 (mod 6), a(p^e) = (-1)^e if p == 5 (mod 6).
a(2*n) = a(3*n) = 0 unless n=0, a(6*n + 5) = -1, a(6*n + 1) = 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).
a(n) = A134667(n), n>0. - R. J. Mathar, Aug 05 2009
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

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A163812 Expansion of (1 - x^5) * (1 - x^6) / ((1 - x) * (1 - x^10)) in powers of x.

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A163817 Expansion of (1 - x^2) * (1 - x^5) / ((1 - x) * (1 - x^6)) in powers of x.

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