[seqfan] Re: A269254

[Don Reble]

> 123 ... remains an obstacle to a significantly larger b-file.

    The sequence for A269254(123) has a factorization pattern.
    For 1 < k < 25000: s(k) is divisible by Lucas(2k+1), unless
    Lucas(2k+1) is divisible by 11 (unless 5 divides 2k+1),
    whence s(k) is divisible by Lucas(2k+1)/11.

    (Here, Lucas() is the additive sequence A000032,
     L(0)=2,1,3,4,7,11,18,...)


    It may have something to do with these rarefying polynomials:

s[k] =                 2 s[k-0] - s[k-0]
s[k] =                 n s[k-1] - s[k-2]
s[k] =         (n^2 - 2) s[k-2] - s[k-4]
s[k] =        (n^3 - 3n) s[k-3] - s[k-6]
s[k] =  (n^4 - 4n^2 + 2) s[k-4] - s[k-8]
s[k] = (n^5 - 5n^3 + 5n) s[k-5] - s[k-10]
s[k] =              P[j] s[k-j] - s[k-2j]

    P[j] = n P[j-1] - P[j-2]

    110, 488, 702, 1298, and 2158 are values of (n^2-2),
    and 123 is a value of (n^5-5n^3+5n).

-- 
Don Reble  djr at nk.ca