A194880 The numerators of the inverse Akiyama-Tanigawa algorithm from A001045(n).

0, -1, -1, -4, -5, -2, -7, -8, -3, -10, -11, -4, -13, -14, -5, -16, -17, -6, -19, -20, -7, -22, -23, -8, -25, -26, -9, -28, -29, -10, -31, -32, -11, -34, -35, -12, -37, -38, -13, -40, -41, -14, -43, -44, -15, -46, -47, -16, -49, -50, -17, -52, -53, -18, -55, -56, -19, -58, -59, -20

Offset: 0

Author

Paul Curtz, Sep 07 2011

Keywords

• sign
• tabl
• frac

Comments

0, -1, -1, -4/3, -5/3, -2, -7/3, -8/3, -3, -10/3, -11/3, -4, -13/4, -14/3, -5, = a(n)/b(n),

1, 0, 1, 4/3, 5/3, 2, 7/3, 8/3, 3,

1, -2, -1, -4/3, -5/3, -2, -7/3, -8/3, -3,

3, -2, 1, 4/3, 5/3, 2, 7/3, 8/3, 3,

5, -6, -1, -4/3, -5/3, -2, -7/3, -8/3, -3,

11, -10, 1, 4/3, 5/3, 2, 7/3, 8/3, 3,

21, -22, -1, -4/3, -5/3, -2, -7/3, -8/3, -3,

Vertical: A001045(n), -A078008(n), (-1)^(n+1)*A000012(n), (-1)^(n+1)*A010709(n)/A010701(n), (-1)^(n+1)*A010716(n+1)/A010701(n), A007395(n), .. .

a(n)=0, 1 before (-A145064(n+1)=-A051176(n+3).

b(n)=1, 1 before A169609(n). b(n)=1, 1, 1 before A144437(n+1).

a(n+5)-a(n+2)=b(n+5) (=-1,-3,-3,=-A169609(n)).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• D. Merlini, R. Sprugnoli, and M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.
• Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Programs

Mathematica

a[0]=0; a[1]=-1; a[n_] := (-n-1)/Max[1, 2*Mod[n, 3]-1]; Table[a[n], {n, 0, 59}] (* Jean-François Alcover, Sep 18 2012 *)

Formula

a(3*n)=-3*n-1 except a(0)=0; a(3*n+1)=-3*n-2 except a(1)=-1; a(3*n+2)=-n-1.

From Chai Wah Wu, May 07 2024: (Start)

a(n) = 2*a(n-3) - a(n-6) for n > 7.

G.f.: x*(x^6 + x^5 - 3*x^3 - 4*x^2 - x - 1)/(x^6 - 2*x^3 + 1). (End)