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A163818 Expansion of (1 - x) * (1 - x^6) / ((1 - x^2) * (1 - x^5)) in powers of x. 3

%I #20 Oct 02 2023 11:32:21

%S 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,

%T -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,

%U -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

%N Expansion of (1 - x) * (1 - x^6) / ((1 - x^2) * (1 - x^5)) in powers of x.

%H G. C. Greubel, <a href="/A163818/b163818.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-1, -1, -1, -1).

%F Euler transform of length 6 sequence [-1, 1, 0, 0, 1, -1].

%F 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.

%F a(n) = -a(-n) unless n=0. a(n+5) = a(n) unless n=0 or n=-5.

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

%F G.f.: 1 / (1 + x / (1 + x^4 / (1 + x^2))). - _Michael Somos_, Jan 03 2013

%F a(n) = (-1)^n * A163812(n). Convolution inverse of A163817.

%e G.f. = 1 - x + x^2 - x^3 + x^4 - x^6 + x^7 - x^8 + x^9 - x^11 + x^12 - x^13 + ...

%t a[ n_] := Boole[n == 0] + {-1, 1, -1, 1, 0}[[Mod[n, 5, 1]]]; (* _Michael Somos_, Jun 17 2015 *)

%t a[ n_] := Boole[n == 0] + (-1)^(n + Quotient[n, 5]) Sign@Mod[n, 5]; (* _Michael Somos_, Jun 17 2015 *)

%t CoefficientList[Series[(1 + x^2 + x^4) / (1 + x + x^2 + x^3 + x^4), {x, 0, 100}], x] (* _Vincenzo Librandi_, Aug 05 2017 *)

%o (PARI) {a(n) = (n==0) + [ 0, -1, 1, -1, 1][n%5 + 1]};

%o (PARI) {a(n) = (n==0) + (-1)^(n + n\5) * kronecker(25, n)};

%Y Cf. A163813, A163817.

%K sign,easy

%O 0,1

%A _Michael Somos_, Aug 04 2009

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Last modified July 15 23:38 EDT 2024. Contains 374343 sequences. (Running on oeis4.)