A168505 -id:A168505 - cp's OEIS frontend

A257075 a(n) = (-1)^(n mod 3).

1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1

Offset: 0

Michael Somos, Apr 15 2015

Period 3: repeat [1, -1, 1]. - Wesley Ivan Hurt, Jul 02 2016

			G.f. = 1 - x + x^2 + x^3 - x^4 + x^5 + x^6 - x^7 + x^8 + x^9 - x^10 + ...
G.f. = q - q^3 + q^5 + q^7 - q^9 + q^11 + q^13 - q^15 + q^17 + q^19 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A008611, A086953, A130151, A168505, A257076.

Essentially the same as A131561.

[(-1)^(n mod 3) : n in [0..100]]; // Wesley Ivan Hurt, Jul 02 2016

A257075:=n->(-1)^(n mod 3): seq(A257075(n), n=0..100); # Wesley Ivan Hurt, Jul 02 2016

a[ n_] := (-1)^Mod[n, 3]; Table[a[n], {n, 0, 100}]
LinearRecurrence[{0,0,1},{1,-1,1},80] (* or *) PadRight[{},100,{1,-1,1}] (* Harvey P. Dale, May 25 2023 *)

PARI
```
{a(n) = (-1)^(n%3)};
```
PARI
```
{a(n) = 1 - 2 * (n%3 == 1)};
```
PARI
```
{a(n) = [1, -1, 1][n%3 + 1]};
```

{a(n) = my(A, p, e); n = abs(2*n + 1); A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p==3, -1, 1))};

Euler transform of length 6 sequence [-1, 1, 2, 0, 0, -1].
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = -1 if e>0, otherwise b(p^e) = 1.
a(n) = a(-1-n) = a(n+3) = -a(n-1)*a(n-2) for all n in Z.
G.f.: (1 - x + x^2) / (1 - x^3).
G.f.: (1 - x) * (1 - x^6) / ((1 - x^2) * (1 - x^3)^2).
G.f.: 1 / (1 + x / (1 + 2*x^2 / (1 - x / (1 - x / (1 + x))))).
Given g.f. A(x), then x*A(x^2) = Sum_{k>0} (x^k - x^(2*k)) - 2*(x^(3*k) - x^(6*k)).
a(n) = A131561(n+1) for all n in Z.
a(n) = (-1)^n * A130151(n) for all n in Z.
Convolution inverse is A257076.
PSUM transform is A008611.
BINOMIAL transform is A086953.
1 / (1 - a(0)*x / (1 - a(1)*x / (1 - a(2)*x / ...))) is the g.f. of A168505.
From Wesley Ivan Hurt, Jul 02 2016: (Start)
a(n) = (1 + 2*cos(2*n*Pi/3) - 2*sqrt(3)*sin(2*n*Pi/3))/3.
a(n) = 2*sgn((n+2) mod 3) - 1. (End)
E.g.f.: (exp(3*x/2) + 4*sin(Pi/6-sqrt(3)*x/2))*exp(-x/2)/3. - Ilya Gutkovskiy, Jul 02 2016

A198379 Triangle T(n,k), read by rows, given by (0,1,-1,0,1,-1,0,1,-1,0,1,-1,...) DELTA (1,0,0,1,0,0,1,0,0,1,0,0,1,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 0, 3, 1, 0, 0, -1, 0, 4, 1, 0, 0, 0, -4, 0, 5, 1, 0, 0, 0, 0, -10, 0, 6, 1, 0, 0, 0, 2, 0, -20, 0, 7, 1, 0, 0, 0, 0, 12, 0, -35, 0, 8, 1, 0, 0, 0, 0, 0, 42, 0, -56, 0, 9, 1

Offset: 0

Philippe Deléham, Oct 28 2011

Equal to A174014*A130595 as infinite lower triangular matrices.

			Triangle begins :
1
0, 1
0, 1, 1
0, 0, 2, 1
0, 0, 0, 3, 1
0, 0, -1, 0, 4, 1
0, 0, 0, -4, 0, 5, 1
0, 0, 0, 0, -10, 0, 6, 1

Cf. A174014, A174015, A174016

Sum_k>=0 T(n,k)= A174015(n).
Sum_k>=0 T(n,k)*2^k = A174016(n).
Sum_ {0<=k<=n} T(n,k)*(-1)^(n-k) = A168505(n).

cp's OEIS Frontend

A257075 a(n) = (-1)^(n mod 3).

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A198379 Triangle T(n,k), read by rows, given by (0,1,-1,0,1,-1,0,1,-1,0,1,-1,...) DELTA (1,0,0,1,0,0,1,0,0,1,0,0,1,...) where DELTA is the operator defined in A084938.

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