A257179 -id:A257179 - cp's OEIS frontend

A259042 Period 8 sequence [0, 1, 1, 1, 2, 1, 1, 1, ...].

0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1

Offset: 0

Michael Somos, Jun 17 2015

			G.f. = x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + x^7 + x^9 + x^10 + x^11 + ...

Antti Karttunen, Table of n, a(n) for n = 0..65537
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (1,0,0,-1,1).

Cf. A010877, A257179.

a[ n_] := {1, 1, 1, 2, 1, 1, 1, 0}[[Mod[n, 8, 1]]];
a[ n_] := SeriesCoefficient[ 1 / (1 - x) - 1 / (1 + x^4), {x, 0, Abs@n}];

{a(n) = 1 + (n%4 == 0) - 2*(n%8 == 0)};

{a(n) = [ 0, 1, 1, 1, 2, 1, 1, 1][n%8 + 1]};

{a(n) = polcoeff( 1 / (1 - x) - 1 / (1 + x^4) + x * O(x^abs(n)), abs(n))};

Euler transform of length 8 sequence [1, 0, 1, -1, 0, -1, 0, 1].
Moebius transform is length 8 sequence [1, 0, 0, 1, 0, 0, 0, -2].
a(n) is multiplicative with a(2) = 1, a(4) = 2, a(2^e) = 0 if e > 2, a(p^e) = 1 if p > 2.
G.f.: x * (1 + x^3)/((1-x)*(1 + x^4)).
G.f.: x * (1 - x^4)*(1 - x^6)/((1-x)*(1 - x^3)*(1 - x^8)).
G.f.: 1/(1-x) - 1/(1 + x^4).
a(n) = a(-n) = a(n+8) for all n in Z.
a(2*n + 1) = a(4*n + 2) = 1.  a(8*n) = 0. a(8*n + 4) = 2.
a(n) = A257179(n+4) unless n = -4.
Dirichlet g.f.: zeta(s) * (1 + 4^(-s) - 2 * 8^(-s)). - Álvar Ibeas, Mar 18 2021

More terms from Antti Karttunen, Jul 29 2018

A257181 Expansion of (1 - x) * (1 + x^4) / (1 + x^5) in powers of x.

Original entry on oeis.org

1, -1, 0, 0, 1, -2, 1, 0, 0, -1, 2, -1, 0, 0, 1, -2, 1, 0, 0, -1, 2, -1, 0, 0, 1, -2, 1, 0, 0, -1, 2, -1, 0, 0, 1, -2, 1, 0, 0, -1, 2, -1, 0, 0, 1, -2, 1, 0, 0, -1, 2, -1, 0, 0, 1, -2, 1, 0, 0, -1, 2, -1, 0, 0, 1, -2, 1, 0, 0, -1, 2, -1, 0, 0, 1, -2, 1, 0, 0

Offset: 0

Michael Somos, Apr 17 2015

			G.f. = 1 - x + x^4 - 2*x^5 + x^6 - x^9 + 2*x^10 - x^11 + x^14 - 2*x^15 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,-1).

Cf. A164116, A257179.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x)*(1+x^4)/(1+x^5))); // G. C. Greubel, Aug 02 2018

a[ n_] := -Boole[n == 0] + {-1, 0, 0, 1, -2, 1, 0, 0, -1, 2}[[Mod[n, 10, 1]]];
a[ n_] := SeriesCoefficient[ (1 - x) * (1 + x^4) / (1 + x^5), {x, 0, Abs@n}];
CoefficientList[Series[(1-x)*(1+x^4)/(1+x^5), {x, 0, nmax}], x] (* G. C. Greubel, Aug 02 2018 *)

{a(n) = if( n==0, 1, (-1)^(n\5) * [2, -1, 0, 0, 1][n%5 + 1])};

{a(n) = polcoeff( (1 - x) * (1 + x^4) / (1 + x^5) + x * O(x^abs(n)), abs(n))};

x='x+O('x^60); Vec((1-x)*(1+x^4)/(1+x^5)) \\ G. C. Greubel, Aug 02 2018

Euler transform of length 10 sequence [-1, 0, 0, 1, -1, 0, 0, -1, 0, 1].
a(n) = a(-n) for all n in Z. a(n+5) = -a(n) unless n = 0 or -5. a(5*n) = 2 * (-1)^n unless n = 0. a(5*n + 2) = a(5*n + 3) = 0. a(5*n + 1) = a(5*n - 1) = -(-1)^n.
G.f.: (1 - x) * (1 + x^4) / (1 + x^5).
G.f.: (1 - x) * (1 - x^5) * (1 - x^8) / ((1 - x^4) * (1 - x^10)).
Convolution inverse is A257179.
a(n) = (-1)^floor( (n+4) / 5) * A164116(n).

A323191 Expansion of (1 - x^5) / ((1 + x) * (1 + x^4)) in powers of x.

Original entry on oeis.org

1, -1, 1, -1, 0, -1, 1, -1, 2, -1, 1, -1, 0, -1, 1, -1, 2, -1, 1, -1, 0, -1, 1, -1, 2, -1, 1, -1, 0, -1, 1, -1, 2, -1, 1, -1, 0, -1, 1, -1, 2, -1, 1, -1, 0, -1, 1, -1, 2, -1, 1, -1, 0, -1, 1, -1, 2, -1, 1, -1, 0, -1, 1, -1, 2, -1, 1, -1, 0, -1, 1, -1, 2, -1, 1

Offset: 0

Michael Somos, Jan 06 2019

			G.f. = 1 - x + x^2 - x^3 - x^5 + x^6 - x^7 + 2*x^8 - x^9 + x^10 + ...

Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (-1,0,0,-1,-1).

Cf. A257179.

a[ n_] := (-1)^n + If[Mod[n, 4] == 0, (-1)^(n/4), 0] - Boole[n == 0];
a[ n_] := {-1, 1, -1, 0, -1, 1, -1, 2}[[Mod[n, 8, 1]]] - Boole[n == 0];
a[ n_] := SeriesCoefficient[ (1 - x^5) / ((1 + x) * (1 + x^4)), {x, 0, Abs@n}];

{a(n) = (-1)^n + if(n%4==0, (-1)^(n/4)) - (n==0)};

{a(n) = [2, -1, 1, -1, 0, -1, 1, -1][n%8 + 1] - (n==0)};

{a(n) = n=abs(n); polcoeff(  (1 - x^5) / ((1 + x) * (1 + x^4)) + x * O(x^n), n)};

{a(n) = my(e); n=abs(n); if( n<1, n==0, e=valuation(n, 2); -if( e<3, [1, -1, 0][e+1], -2))};

a(n) = -b(n) and b() is multiplicative with b(2) = -1, b(4) = 0, b(2^e) = -2 if e>2, b(p^e) = 1 if p>2.
Euler transform of length 8 sequence [-1, 1, 0, -1, -1, 0, 0, 1].
Moebius transform is length 8 sequence [-1, 2, 0, -1, 0, 0, 0, 2].
G.f.:  (1 - x^5) / ((1 + x) * (1+ x^4)) = -1 + 1 / (1 + x) + 1 / (1 + x^4).
a(n) = a(-n) for all n in Z. a(n+4) = a(n-4) except if n=4 or n=-4.
a(n) = (-1)^n * A257179(n), a(2*n + 1) = -1, a(4*n + 2) = 1, a(8*n + 4) = 0 for all n in Z.

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A259042 Period 8 sequence [0, 1, 1, 1, 2, 1, 1, 1, ...].

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A257181 Expansion of (1 - x) * (1 + x^4) / (1 + x^5) in powers of x.

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A323191 Expansion of (1 - x^5) / ((1 + x) * (1 + x^4)) in powers of x.

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