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A242062 Expansion of x * (1 - x^12) / ((1 - x^3) * (1 - x^4) * (1 - x^7)) in powers of x.

%I A242062 #21 Sep 29 2024 13:39:54
%S A242062 0,1,0,0,1,1,0,1,2,1,1,2,2,1,2,3,2,2,3,3,2,3,4,3,3,4,4,3,4,5,4,4,5,5,
%T A242062 4,5,6,5,5,6,6,5,6,7,6,6,7,7,6,7,8,7,7,8,8,7,8,9,8,8,9,9,8,9,10,9,9,
%U A242062 10,10,9,10,11,10,10,11,11,10,11,12,11,11,12
%N A242062 Expansion of x * (1 - x^12) / ((1 - x^3) * (1 - x^4) * (1 - x^7)) in powers of x.
%H A242062 Vincenzo Librandi, <a href="/A242062/b242062.txt">Table of n, a(n) for n = 0..5000</a>
%F A242062 Euler transform of length 12 sequence [0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, -1].
%F A242062 G.f.: x * (1 - x + x^3 - x^5 + x^6) / (1 - x - x^7 + x^8).
%F A242062 G.f.: x / (1 - x^3 / (1 - x / (1 + x / (1 - x^5 / (1 - x / (1 + x^2 / (1 - x^2))))))).
%F A242062 a(n) = -a(-n) = a(n-7) + 1 = a(n-1) + a(n-7) - a(n-8) for all n in Z.
%F A242062 0 = 2*a(n) - a(n+1) + a(n+2) - 2*a(n+3) + (a(n+1) - a(n+2))^2 for all n in Z.
%F A242062 a(n+1) - a(n) = A131372(n).
%e A242062 G.f. = x + x^4 + x^5 + x^7 + 2*x^8 + x^9 + x^10 + 2*x^11 + 2*x^12 + x^13 + ...
%p A242062 a:= n -> [0, 1, 0, 0, 1, 1, 0][n mod 7 + 1] + floor(n/7):
%p A242062 seq(a(n), n=0..20); # _Robert Israel_, Aug 13 2014
%t A242062 a[ n_] := Quotient[ n+3, 7] + {1, 0, 0, 0, 0, -1, 0}[[Mod[ n, 7, 1]]];
%t A242062 a[ n_] := Sign[n] * SeriesCoefficient[ x * (1 - x^12) / ((1 - x^3) * (1 - x^4) * (1 - x^7)), {x, 0, Abs[n]}];
%t A242062 CoefficientList[Series[x (1 - x + x^3 - x^5 + x^6)/(1 - x - x^7 + x^8), {x, 0, 100}], x] (* _Vincenzo Librandi_, Aug 14 2014 *)
%t A242062 LinearRecurrence[{1,0,0,0,0,0,1,-1},{0,1,0,0,1,1,0,1},90] (* _Harvey P. Dale_, Sep 29 2024 *)
%o A242062 (PARI) {a(n) = (n+3)\7 + (n%7==1) - (n%7==6)};
%o A242062 (PARI) {a(n) = sign(n) * polcoeff( x * (1 - x^12) / ((1 - x^3) * (1 - x^4) * (1 - x^7)) + x * O(x^abs(n)), abs(n))};
%o A242062 (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1 -x+x^3-x^5+x^6)/(1-x-x^7+x^8))); // _G. C. Greubel_, Aug 05 2018
%Y A242062 Cf. A131372.
%K A242062 nonn
%O A242062 0,9
%A A242062 _Michael Somos_, Aug 13 2014