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Values of n for which there exist d(1),...,d(n), each in {0,1,...,4} and an r in {1,...,4} such that Sum[d(i)d(i+k),i=1,n-k]=r (mod 5) for all k=0,...,n-1. (Such a sequence is called a very(5,r) sequence. See the link.).
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%I #12 Aug 16 2020 10:25:09

%S 1,3,6,10,13,16

%N Values of n for which there exist d(1),...,d(n), each in {0,1,...,4} and an r in {1,...,4} such that Sum[d(i)d(i+k),i=1,n-k]=r (mod 5) for all k=0,...,n-1. (Such a sequence is called a very(5,r) sequence. See the link.).

%C Conjecture. Let A be a very(5,1) (respectively very(5,4)) sequence of length n and let Z be a sequence of n-1 0's.. Then AZ(3A)ZA is a very(5,1) (respectively very(5,4)) sequence of length 5n-2. (Here 3A denotes the result of multiplying each term of A by 3, then reducing modulo 5; and juxtaposition of symbols denotes concatenation of sequences.)

%H John W. Layman, <a href="http://intranet.math.vt.edu/people/layman/sequences/very_br.htm">On A Generalization of Very Odd Sequences</a>

%Y Cf. A053006, A117548.

%K nonn,more

%O 1,2

%A _John W. Layman_, Apr 21 2006