[seqfan] From Balmer

[Paul Curtz]

Hello,

Consider

0, 5/36, 21/100, 45/196, 117/484, 165/676, 285/1156, 357/1444, 525/2116, 837/3364, 357/3844, ... =  f(n+2)/g(n+2).
  
Numerators: A166010 = A061037(A000040).

Denominators:
a)      g1(n+2)  =     A061038(A000040)          if A061038(2) =  1 
b)      g2(n+2)  =     A069262 = A100484^2       if A061038(2) = 16.
  g2(n+2) = 4*A166010 + 16  (A010855).

(First differences:
5/36, 16/225, 24/1225, 72/5929, 48/20449, ... = A069482/A166329.)

1) g(n+3) - f(n+3) and its first differences:

    31, 79, 151, 367, 511, 871, 1087, 1591, ... = a(n) 
    48, 72, 216, 144, 360, 216,  504,  936, ... = b(n) = 24*A267896.

a(n) - 1 = 30, 78, 150, ... = 6*(A066885= 5, 13, 25, 61, ... which first differences are 4*A267896).

2) g1(n+2) + f(n+2) and its first differences:

 1, 41, 121, 241, 601, 841, 1441, 1801, 2641, ... = c(n)
40, 80, 120, 360, 240, 600,  360,  840, 1560, ... = d(n) = 40*A267896.

c(n) - 1 = 0, 40, 120, 240, 600, ... = 0 followed by 40*A061066.

Paul