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A164115 Expansion of (1 - x^5) / ((1 - x) * (1 - x^4)) in powers of x.

%I A164115 #35 Feb 19 2024 01:50:06
%S A164115 1,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,
%T A164115 1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,
%U A164115 2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2
%N A164115 Expansion of (1 - x^5) / ((1 - x) * (1 - x^4)) in powers of x.
%C A164115 The sequence A107453 has the same terms but different offset.
%C A164115 Convolution inverse of A164116.
%C A164115 Decimal expansion of 11111/99990. - _Elmo R. Oliveira_, Feb 18 2024
%H A164115 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,1).
%F A164115 Euler transform of length-5 sequence [ 1, 0, 0, 1, -1].
%F A164115 a(n) is multiplicative with a(2) = 1, a(2^e) = 2 if e>1, a(p^e) = 1 if p>2.
%F A164115 a(n) = (-1)^n * A164117(n).
%F A164115 a(4*n) = 2 unless n=0. a(2*n + 1) = a(4*n + 2) = 1.
%F A164115 a(-n) = a(n). a(n+4) = a(n) unless n=0 or n=-4.
%F A164115 G.f.: (1 + x + x^2 + x^3 + x^4) / ((1+x)*(1-x)*(1+x^2)).
%F A164115 a(n) = A138191(n+2), n>0. - _R. J. Mathar_, Aug 17 2009
%F A164115 Dirichlet g.f. (1+1/4^s)*zeta(s). - _R. J. Mathar_, Feb 24 2011
%F A164115 a(n) = (i^n + (-i)^n + (-1)^n + 5)/4 for n > 0 where i is the imaginary unit. - _Bruno Berselli_, Feb 25 2011
%e A164115 1 + x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + x^7 + 2*x^8 + x^9 + x^10 + ...
%t A164115 CoefficientList[Series[(1+x+x^2+x^3+x^4)/(1-x^4), {x, 0, 100}], x] (* _G. C. Greubel_, Sep 22 2018 *)
%t A164115 LinearRecurrence[{0,0,0,1},{1,1,1,1,2},120] (* or *) PadRight[{1},120,{2,1,1,1}] (* _Harvey P. Dale_, Aug 24 2019 *)
%o A164115 (PARI) {a(n) = 2 - (n==0) - (n%4>0)}
%o A164115 (PARI) x='x+O('x^99); Vec((1-x^5)/((1-x)*(1-x^4))) \\ _Altug Alkan_, Sep 23 2018
%o A164115 (Magma) m:=100; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x+x^2+x^3+x^4)/(1-x^4))); // _G. C. Greubel_, Sep 22 2018
%Y A164115 Cf. A107453, A138191, A164116, A164117.
%K A164115 nonn,mult,easy
%O A164115 0,5
%A A164115 _Michael Somos_, Aug 10 2009