A106510 Expansion of (1+x)^2/(1+x+x^2).
1, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1
Offset: 0
Examples
1 + x - x^2 + x^4 - x^5 + x^7 - x^8 + x^10 - x^11 + x^13 - x^14 + ...
Links
- Index entries for linear recurrences with constant coefficients, signature (-1,-1).
Programs
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Mathematica
{1}~Join~LinearRecurrence[{-1, -1}, {1, -1}, 105] (* Jean-François Alcover, Oct 28 2019 *) -
PARI
{a(n) = if( n==0, 1, [0, 1, -1][n%3 + 1])} \\ Michael Somos, Oct 15 2008 -
PARI
{a(n) = if( n==0, 1, kronecker(-3, n))} \\ Michael Somos, Oct 15 2008 -
PARI
A106510(n)=kronecker(-3, n+!n) \\ M. F. Hasler, May 07 2018
Formula
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} (-1)^j*binomial(2n-k-j, j)
From Michael Somos, Oct 15 2008: (Start)
Euler transform of length 3 sequence [ 1, -2, 1].
a(n) is multiplicative with a(3^e) = 0^e, a(p^e) = 1 if p == 1 (mod 3), a(p^e) = (-1)^e if p == 2 (mod 3).
G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v) = 4 - 3*v - u * (4 - 2*v - u). (End)
a(-n) = a(n). a(n+3) = a(n) unless n = 0 or n = -3.
a(n) = Sum_{k=0..n} A128908(n,k)*(-1)^(n-k). - Philippe Deléham, Jan 22 2012
Extensions
Edited by M. F. Hasler, May 07 2018
Comments