A105368 -id:A105368 - cp's OEIS frontend

A105367 Expansion of (1-x^3)/(1-x^5).

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

Offset: 0

Paul Barry, Apr 01 2005

Periodic {1,0,0,-1,0}. Partial sums of A105368. Binomial transform of A105370.

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

G.f.: (1+x+x^2)/(1+x+x^2+x^3+x^4);

a(n) = sqrt(2/5 - 2*sqrt(5)/25)*cos(4*Pi*n/5 + 3*Pi/10) + sqrt(2/5 + 2*sqrt(5)/25)*sin(2*Pi*n/5 + 2*Pi/5).

A138019 Period 5: repeat [1, 1, 0, -1, -1].

Original entry on oeis.org

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

Offset: 0

Paul Curtz, May 01 2008

			G.f. = 1 + x - x^3 - x^4 + x^5 + x^6 - x^8 - x^9 + x^10 + x^11 - x^13 + ...

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

Cf. A080891, A105368.

A138019 := proc(n)
    op( 1+modp(n,5),[1,1,0,-1,-1]) ;
end proc:
seq(A138019(n),n=0..30) ; # R. J. Mathar, Feb 12 2021

a[ n_] := Sign[2 - Mod[n, 5]]; (* Michael Somos, Jun 17 2015 *)
PadRight[{},120,{1,1,0,-1,-1}] (* Harvey P. Dale, Oct 06 2023 *)

a(n)=sign(2-n%5) /* Jaume Oliver Lafont, Aug 28 2009 */

Inverse binomial transform of A138003.
O.g.f.: (1+x)(x^2+x+1)/(1+x+x^2+x^3+x^4). - R. J. Mathar, Jun 28 2008
Euler transform of length 5 sequence [ 1, -1, -1, 0, 1]. - Michael Somos, Jun 17 2015
G.f.: (1 - x^2 ) * (1 - x^3) / ((1 - x) * (1 - x^5)). - Michael Somos, Jun 17 2015
a(n) = -a(-1-n) = a(n+5) for all n in Z. - Michael Somos, Jun 17 2015

A259030 a(n) is multiplicative with a(2^e) = -(1 - (-1)^e) / 2 if e > 0, a(p^e) = Kronecker(5, p)^e if p > 2.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Jun 17 2015

			G.f. = x - x^2 - x^3 + x^6 - x^7 - x^8 + x^9 + x^11 - x^13 + x^14 - x^17 + ...

Cf. A080891, A105368, A113185.

f[p_, e_] := KroneckerSymbol[5, p]^e; f[2, e_] := -(1 - (-1)^e) / 2; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 04 2023 *)

{a(n) = my(A, p, e); if( !n, 0, A = factor(abs(n)); prod(k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, -(e%2), kronecker( 5, p)^e)))};

a(n) = a(-n) for all n in Z.
a(2*n + 1) = A105368(n). a(4*n + 1) = A080891(n-1). a(4*n + 2) = - A105368(n). a(4*n - 1) = A080891(n+1).
A113185(n) = Sum_{d|n} d * a(d) * -(-1)^(n/d) if n > 0.
G.f.: f(x) - Sum_{k>0} f(x^2^(2*k-1)) where f(x) := x * (1 - x^2) * (1 - x^6) / (1 - x^10).

cp's OEIS Frontend

A105367 Expansion of (1-x^3)/(1-x^5).

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A138019 Period 5: repeat [1, 1, 0, -1, -1].

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A259030 a(n) is multiplicative with a(2^e) = -(1 - (-1)^e) / 2 if e > 0, a(p^e) = Kronecker(5, p)^e if p > 2.

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