A183912 - OEIS

%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