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%I #10 Nov 23 2023 16:12:23
%S 0,7,16,29,52,82,122,182,259,363,492,648,816,1018,1268,1586,1973,2419,
%T 2904,3452,4063,4762,5543,6421,7393,8487,9700,11052,12543,14183,15960,
%U 17915,20023,22303,24760,27422,30279,33373,36697,40284,44131,48250,52614
%N Number of nondecreasing sequences of n 1..6 integers with no element dividing the sequence sum.
%H R. H. Hardin, <a href="/A212866/b212866.txt">Table of n, a(n) for n = 1..210</a>
%F Empirical: a(n) = 3*a(n-1) -2*a(n-2) -2*a(n-3) +2*a(n-4) +a(n-5) +a(n-6) -4*a(n-7) +4*a(n-9) -a(n-10) -a(n-11) -a(n-12) -a(n-13) +4*a(n-14) -4*a(n-16) +a(n-17) +a(n-18) +2*a(n-19) -a(n-20) -5*a(n-21) +5*a(n-22) +a(n-23) -2*a(n-24) -a(n-25) -a(n-26) +4*a(n-27) -4*a(n-29) +a(n-30) +a(n-31) +a(n-32) +a(n-33) -4*a(n-34) +4*a(n-36) -a(n-37) -a(n-38) -2*a(n-39) +2*a(n-40) +2*a(n-41) -3*a(n-42) +a(n-43).
%e Some solutions for n=8:
%e ..2....3....2....2....2....2....3....3....2....2....2....2....4....5....3....2
%e ..2....3....2....2....3....5....3....4....2....3....2....2....4....5....3....3
%e ..2....3....3....2....3....6....4....4....3....4....2....3....4....5....3....3
%e ..3....3....3....2....3....6....4....4....3....4....2....3....5....5....3....3
%e ..3....3....4....2....6....6....4....5....4....4....2....4....6....6....3....3
%e ..3....3....5....2....6....6....4....6....4....4....2....5....6....6....5....3
%e ..4....3....6....4....6....6....4....6....5....5....2....5....6....6....6....3
%e ..4....4....6....5....6....6....5....6....6....5....5....5....6....6....6....3
%p S6:= combinat:-powerset({$2..6}):
%p f:= proc(n) local s,t,G,S,i,j,T;
%p t:= 0:
%p for S in S6 do
%p G:= coeff(mul(add(x^i*y^(i*j),i=0..n),j=S),x,n);
%p T:= select(s -> S = select(k -> s mod k <> 0, {$2..6}), [$2*n..6*n]);
%p t:= t + add(coeff(G,y,s),s= T);
%p od;
%p t
%p end proc:
%p map(f, [$1..50]); # _Robert Israel_, Nov 23 2023
%Y Column 6 of A212868.
%K nonn
%O 1,2
%A _R. H. Hardin_, May 29 2012
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