A163812 -id:A163812 - 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.

A330025 a(n) = (-1)^floor(n/5) * sign(mod(n, 5)).

Original entry on oeis.org

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

Offset: 0

Michael Somos, Nov 27 2019

This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = 1, z = 1. - Michael Somos, Mar 17 2020

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

Clark Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1).

Cf. A011558, A099443, A163812, A332828.

a[ n_] := (-1)^Quotient[n, 5] Sign@Mod[n, 5];

PARI
```
{a(n) = (-1)^(n\5) * sign(n%5)};
```

Euler transform of length 10 sequence [1, 0, 0, -1, -1, 0, 0, 0, 0, 1].
G.f.: x * (1 + x) * (1 + x^2) / (1 + x^5).
a(n) = A099443(n-1). a(n) = A163812(n) except n=0.
a(n) = (-1)^floor(n/5) * A011558(n) for all n in Z.
0 = a(n)*a(n+4) - a(n+1)*a(n+3) + a(n+2)^2 = a(n)*a(n+5) - a(n+1)*a(n+4) + a(n+2)*a(n+3) for all n in Z. - Michael Somos, Mar 17 2020

cp's OEIS Frontend

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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A330025 a(n) = (-1)^floor(n/5) * sign(mod(n, 5)).

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