Integers of the form (1+2x+4x^3)/(x+n)

[zak seidov]

Not submitted yet.
 
My Qs to gurus:

No solutions for n>10?
Any theory behind all this?

Thanks, Zak

%I A000001
%S A000001 4,8,4,4,4,16,8,4,8,4
%N A000001 Number of integers m of the form
(1+2x+4x^3)/(x+n)
%C A000001 No solutions for n>10?
%e A000001 

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 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}.

%O A000001 1
%K A000001 ,nonn,
%A A000001 Zak Seidov  (zakseidov at yahoo.com), Oct 18
2006


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