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A105368 Expansion of (1-x-x^3+x^4)/(1-x^5).

%I A105368 #42 Nov 07 2024 09:54:15
%S A105368 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,
%T A105368 -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,
%U A105368 -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
%N A105368 Expansion of (1-x-x^3+x^4)/(1-x^5).
%C A105368 Period 5: repeat [1,-1,0,-1,1]. Partial sums are A105367.
%C A105368 Binomial transform of A105369. - _R. J. Mathar_, Jul 29 2010
%H A105368 Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/rfmc.html">Rational Function Multiplicative Coefficients</a>
%H A105368 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-1,-1,-1).
%F A105368 G.f.: (1-x)*(1-x^3)/(1-x^5).
%F A105368 a(n) = A105367(n)-A105367(n-1).
%F A105368 a(n) = (1/2-sqrt(5)/10)*cos(4*Pi*n/5)-sqrt(1/2+sqrt(5)/10)*sin(4*Pi*n/5)+(1/2+sqrt(5)/10)*cos(2*Pi*n/5)-sqrt(1/2-sqrt(5)/10)*sin(2*Pi*n/5).
%F A105368 From _R. J. Mathar_, Jul 29 2010: (Start)
%F A105368 a(n) = -a(n-1) -a(n-2) -a(n-3) -a(n-4), n>4.
%F A105368 G.f.: (1-x)*(1+x+x^2)/(1+x+x^2+x^3+x^4). (End)
%F A105368 a(n) = -sign( (n+3) mod 5)*(-1)^floor(-2n/5-4/5). - _Wesley Ivan Hurt_, Apr 28 2015
%F A105368 a(n) = 2 - ( (n^2+n+1) mod 5 ). - _Wesley Ivan Hurt_, May 31 2015
%F A105368 Euler transform of length 5 sequence [ -1, 0, -1, 0, 1]. - _Michael Somos_, Jun 16 2015
%F A105368 a(n) = a(-1-n) for all n in Z. - _Michael Somos_, Jun 16 2015
%F A105368 a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = b(5^e) = 0^e,  b(p^e) = 1 if p == 1, 9 (mod 10), b(p^e) = (-1)^e if p == 3, 7 (mod 10). - _Michael Somos_, Jun 16 2015
%e A105368 G.f. = 1 - x - x^3 + x^4 + x^5 - x^6 - x^8 + x^9 + x^10 - x^11 - x^13 + ...
%e A105368 G.f. = q - q^3 - q^7 + q^9 + q^11 - q^13 - q^17 + q^19 + q^21 - q^23 + ...
%p A105368 A105368:=n->2-(n^2+n-4 mod 5): seq(A105368(n), n=0..100);
%t A105368 CoefficientList[Series[(1 - x - x^3 + x^4)/(1 - x^5), {x, 0, 100}], x] (* _Vincenzo Librandi_, Apr 29 2015 *)
%t A105368 a[ n_] := {-1, 0, -1, 1, 1}[[Mod[n, 5, 1]]]; (* _Michael Somos_, Jun 16 2015 *)
%o A105368 (Magma) &cat [[1,-1,0,-1,1]: n in [1..20]]; // _Vincenzo Librandi_, Apr 29 2015
%o A105368 (Magma) I:=[1,-1, 0,-1]; [n le 4 select I[n] else -Self(n-1)-Self(n-2)-Self(n-3)-Self(n-4): n in [1..100]]; // _Vincenzo Librandi_, Apr 29 2015
%o A105368 (PARI) {a(n) = [ 1, -1, 0, -1, 1][n%5 + 1]}; /* _Michael Somos_, Jun 16 2015 */
%K A105368 sign,easy
%O A105368 0,1
%A A105368 _Paul Barry_, Apr 01 2005