1,1

The partial fraction decomposition (1+2x+4x^3)/(x+n)=4x^2-4nx+2+4n^2+(1-2n-4n^3)/(x+n) demonstrates that one can generate the solutions for any n by searching through all positive and negative divisors of 1-2n-4n^3, which are set to x+n, such that at least one solution (from the divisor 1, or -1, or 1-2n-4n^3, or -1+2n+4n^3, which may be degenerate) must exist. - R. J. Mathar, Oct 21 2006

Values of m=(1+2x+4x^3)/(x+n):

n, {m_i}

1,{1,35,53,175},

2,{-5,23,73,113,277,419,4109,5791},

3,{-35,263,47117,55255},

4,{-113,509,264245,289495},

5,{-263,875,1006085,1067167},

6,{-509,-1,1,1069,1099,1259,1385,2789,4769,7879,53927,71941,110329,135539,2999933,3125935},

7,{-875,-7,1387,2063,284233,330779,7557149,7789831},

8,{-1385,2933,16826597,17222695},

9,{-2063,-5,2345,4019,657959,748477,34094165,34727695},

10,{-2933,5345,64128365,65092927}.

Corresponding values of x's:

n,{x_i}

1,{0,-2,4,-6},

2,{-1,3,5,-3,-7,-9,33,-37},

3,{-2,-4,110,-116},

4,{-3,-5,259,-267},

5,{-4,-6,504,-514},

6,{-5,-1,1,-11,19,-13,-7,29,-31,-41,119,-131,169,-181,869,-881},

7,{-6,-2,-12,-8,270,-284,1378,-1392},

8,{-7,-9,2055,-2071},

9,{-8,-2,-16,-10,410,-428,2924,-2942},

10,{-9,-11,4009,-4029}.

A122110 := proc(n) local allm, dvs, i, x, m ; allm := {} : dvs := numtheory[divisors](1-2*n-4*n^3) : for i from 1 to nops(dvs) do x := op(i, dvs)-n ; m := (1+2*x+4*x^3)/(x+n) ; allm := allm union {m} ; x := -op(i, dvs)-n ; m := (1+2*x+4*x^3)/(x+n) ; allm := allm union {m} ; od ; RETURN(nops(allm)) ; end : for n from 1 to 100 do printf("%d, ", A122110(n)) ; od ; # R. J. Mathar, Oct 21 2006

Sequence in context: A391987 A045816 A085991 * A394173 A082632 A296481

Adjacent sequences: A122107 A122108 A122109 * A122111 A122112 A122113

Zak Seidov, Oct 18 2006

More terms from R. J. Mathar, Oct 21 2006

approved