[seqfan] Re: A269254
Don Reble
djr at nk.ca
Wed Oct 25 18:38:46 CEST 2017
> 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
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