A113946 Series expansion of Farey rational polynomial based on A112627.

1, 5, 23, 81, 367, 1297, 5871, 20753, 93935, 332049, 1502959, 5312785, 24047343, 85004561, 384757487, 1360072977, 6156119791, 21761167633, 98497916655, 348178682129, 1575966666479, 5570858914065, 25215466663663, 89133742625041

Offset

0,2

Comments

Polynomial expanded is constant*(x+1/2)^2*(1+2x)/(1-x-16x^2+16x^3) the Jasinski rational polynomial p[x_] = (9/32)*(x + 1/2)^3/((x - 1/4)*(x + 1/4)*(x + 1)) f[x_] := 1/p[x] /; 0 <= x <= 1/2 f[x_] := p[x] /; 1/2 < x <= 1 gives a Farey like function with maximum at 1.

Links

Table of n, a(n) for n=0..23.

Index entries for linear recurrences with constant coefficients, signature (-1,16,16).

Formula

b(n) = coefficient series expansion of (9/32)*(x + 1/2)^3/((x - 1/4)*(x + 1/4)*(x + 1)) a(n) = (-16/9)*b(n).

a(n) = (5*(-4)^n+4*(-1)^n+81*4^n)/60 for n>0. G.f.: -(2*x+1)^3 / ((x+1)*(4*x-1)*(4*x+1)). [Colin Barker, Dec 03 2012]

a(n) = -a(n-1)+16*a(n-2)+16*a(n-3). - Wesley Ivan Hurt, May 07 2021

Mathematica

b = -(16/9)*ReplacePart[Table[Coefficient[Series[(9/32)*(x + 1/2)^3/((x - 1/4)*(x + 1/4)*(x + 1)), {x, 0, 30}], x^n], {n, 0, 30}], -9/16, 1]

Crossrefs

Cf. A112627.

Sequence in context: A211922 A139209 A227123 * A283114 A290938 A193696

Adjacent sequences: A113943 A113944 A113945 * A113947 A113948 A113949

Keyword

nonn,easy

Author

Roger L. Bagula, Jan 31 2006

Status

approved