A131372 -id:A131372 - cp's OEIS frontend

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

Michael Somos, Aug 13 2014

nonn

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

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

Cf. A131372.

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

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

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 *)

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

{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))};

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).

A098457 Farey Bisection Expansion of sqrt(7).

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1

Offset: 1

John W. Layman, Sep 08 2004

We define the Farey Bisection Expansion (FBE) of the nonnegative real number x to be the sequence {a(n)} of 0's and 1's determined as follows. Set na(0)=0, da(0)=1, nb(0)=1 and db(0)=0. For n=1, 2, 3,..., set num=na(n-1)+nb(n-1) and den=da(n-1)+db(n-1); if xA010121.

			G.f. = x + x^2 + x^4 + x^6 + x^7 + x^8 + x^9 + x^11 + x^13 + x^14 + x^15 + ...

N. J. A. Sloane, Stern-Brocot or Farey Tree
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,1).

Cf. A010121, A097853, A098458, A131372.

&cat [[1, 1, 0, 1, 0, 1, 1]^^20]; // Wesley Ivan Hurt, Jul 11 2016

seq(op([1, 1, 0, 1, 0, 1, 1]), n=0..20); # Wesley Ivan Hurt, Jul 11 2016

LinearRecurrence[{0, 0, 0, 0, 0, 0, 1},{1, 1, 0, 1, 0, 1, 1},105] (* Ray Chandler, Aug 26 2015 *)

{a(n) = [1, 1, 0, 1, 0, 1, 1][(n-1)%7+1]}; /* Michael Somos, Dec 26 2016 */

From Wesley Ivan Hurt, Jul 11 2016: (Start)
G.f.: x * (1 + x + x^3 + x^5 + x^6) / (1 - x^7).
a(n) = a(n-7) for n>7.
a(n) = 1 - Sum_{k=1..4} floor((n + k)/7)*(-1)^k. (End)
a(n+1) = (-1)^(mod(mod(n, 7), 3)>0) * A131372(n). - Michael Somos, Dec 26 2016

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

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A098457 Farey Bisection Expansion of sqrt(7).

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