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%I #13 Apr 06 2018 03:00:12
%S 2,1,3,2,4,4,1,5,10,5,2,2,17,17,6,1,8,20,38,25,7,2,1,37,66,67,34,8,1,
%T 5,22,124,148,105,44,9,2,4,40,136,309,275,153,55,10,1,4,31,207,470,
%U 637,457,212,67,11,2,1,47,231,778,1193,1163,705,283,80,12,1,10,18,294,1093,2199
%N T(n,k)=Number of nondecreasing arrangements of n+2 numbers in 0..k with each number being the sum mod (k+1) of two others
%C Table starts
%C ..2..1...2....1....2.....1.....2.....1......2......1......2.......1......2
%C ..3..4...5....2....8.....1.....5.....4......4......1.....10.......1......3
%C ..4.10..17...20...37....22....40....31.....47.....18.....63......19.....55
%C ..5.17..38...66..124...136...207...231....294....216....414.....217....430
%C ..6.25..67..148..309...470...778..1093...1504...1636...2521....2217...3249
%C ..7.34.105..275..637..1193..2199..3631...5596...7613..11744...13590..19258
%C ..8.44.153..457.1163..2525..5126..9576..16366..25833..42161...57825..85989
%C ..9.55.212..705.1953..4752.10501.21660..40449..71306.124219..192247.304552
%C .10.67.283.1031.3085..8238.19630.43980..88692.170734.316708..538177.907230
%C .11.80.367.1448.4650.13438.34274.82453.177974.368699.724961.1329686
%C Each column is eventually equal to a polynomial in n (see link). - _Robert Israel_, Apr 05 2018
%H R. H. Hardin, <a href="/A183912/b183912.txt">Table of n, a(n) for n = 1..238</a>
%H Robert Israel, <a href="/A183912/a183912.pdf">Proof of comment</a>
%e All solutions for n=3, k=2
%e ..1....0....0....0....0....0....1....0....0....0
%e ..1....1....0....0....0....1....1....0....0....1
%e ..2....2....1....0....0....1....1....1....0....1
%e ..2....2....1....0....2....1....2....2....1....2
%e ..2....2....2....0....2....2....2....2....1....2
%p k:= 3: N:= 20: # to produce T(n,k) for n=2..N
%p q:= proc(S,x) local L,m,i;
%p m:= nops(S);
%p L:= convert(x+3^m,base,3)[1..m];
%p [seq([S[i],L[i]+1],i=1..m)];
%p end proc:
%p enlarge:= proc(S) local m,j;
%p seq(q(S,j),j=0..3^nops(S)-1)
%p end proc:
%p States:= map(enlarge, combinat:-powerset([$0..k])): ns:= nops(States):
%p T:= Matrix(ns,ns):
%p for j from 1 to ns do
%p S:= States[j];
%p if nops(S)=1 and S[1][2]=1 then T[1,j]:= 1 fi
%p od:
%p for i from 2 to ns do
%p S:= States[i]; P:= S[-1];
%p Sp:= subs(P=[P[1],min(3,P[2]+1)], S);
%p member(Sp, States, 'j');
%p T[i,j]:= 1;
%p for sp from P[1]+1 to k do
%p Sp:= [op(S),[sp,1]];
%p member(Sp, States,'j');
%p T[i,j]:= 1
%p od
%p od:
%p v:= Vector[row]([1,0$(ns-1)]):
%p good:= proc(s) local L:
%p L:= map(p -> p[1]$p[2], States[s]);
%p andmap(j -> member(L[j], [seq(seq(L[i]+L[ip] mod (k+1),ip = {$i+1..nops(L)} minus {j}),i=[$1..j-1,$(j+1)..nops(L)])]),
%p [$1..nops(L)])
%p end proc:
%p goodS:= select(good, [$1..ns]):
%p vT[0]:= v:
%p for i from 1 to N+2 do vT[i]:= vT[i-1] . T od:
%p seq(convert(vT[i][goodS],`+`), i=3..N+2); # _Robert Israel_, Apr 05 2018
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_, Jan 07 2011
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