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

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, 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, 10, 10, 9, 10, 11, 10, 10, 11, 11, 10, 11, 12, 11, 11, 12

Offset: 0

Author

Michael Somos, Aug 13 2014

Keywords

• nonn

Examples

			G.f. = x + x^4 + x^5 + x^7 + 2*x^8 + x^9 + x^10 + 2*x^11 + 2*x^12 + x^13 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A131372.

Programs

Magma

m:=50; R:=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

Maple

a:= n -> [0, 1, 0, 0, 1, 1, 0][n mod 7 + 1] + floor(n/7):
seq(a(n), n=0..20); # Robert Israel, Aug 13 2014

Mathematica

a[ n_] := Quotient[ n+3, 7] + {1, 0, 0, 0, 0, -1, 0}[[Mod[ n, 7, 1]]];
a[ n_] := Sign[n] * SeriesCoefficient[ x * (1 - x^12) / ((1 - x^3) * (1 - x^4) * (1 - x^7)), {x, 0, Abs[n]}];
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 *)
LinearRecurrence[{1,0,0,0,0,0,1,-1},{0,1,0,0,1,1,0,1},90] (* Harvey P. Dale, Sep 29 2024 *)

PARI

{a(n) = (n+3)\7 + (n%7==1) - (n%7==6)};

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))};

Formula

Euler transform of length 12 sequence [0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, -1].

G.f.: x * (1 - x + x^3 - x^5 + x^6) / (1 - x - x^7 + x^8).

G.f.: x / (1 - x^3 / (1 - x / (1 + x / (1 - x^5 / (1 - x / (1 + x^2 / (1 - x^2))))))).

a(n) = -a(-n) = a(n-7) + 1 = a(n-1) + a(n-7) - a(n-8) for all n in Z.

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.

a(n+1) - a(n) = A131372(n).