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A183912 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 8
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, 5, 22, 124, 148, 105, 44, 9, 2, 4, 40, 136, 309, 275, 153, 55, 10, 1, 4, 31, 207, 470, 637, 457, 212, 67, 11, 2, 1, 47, 231, 778, 1193, 1163, 705, 283, 80, 12, 1, 10, 18, 294, 1093, 2199 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Table starts
..2..1...2....1....2.....1.....2.....1......2......1......2.......1......2
..3..4...5....2....8.....1.....5.....4......4......1.....10.......1......3
..4.10..17...20...37....22....40....31.....47.....18.....63......19.....55
..5.17..38...66..124...136...207...231....294....216....414.....217....430
..6.25..67..148..309...470...778..1093...1504...1636...2521....2217...3249
..7.34.105..275..637..1193..2199..3631...5596...7613..11744...13590..19258
..8.44.153..457.1163..2525..5126..9576..16366..25833..42161...57825..85989
..9.55.212..705.1953..4752.10501.21660..40449..71306.124219..192247.304552
.10.67.283.1031.3085..8238.19630.43980..88692.170734.316708..538177.907230
.11.80.367.1448.4650.13438.34274.82453.177974.368699.724961.1329686
Each column is eventually equal to a polynomial in n (see link). - Robert Israel, Apr 05 2018
LINKS
Robert Israel, Proof of comment
EXAMPLE
All solutions for n=3, k=2
..1....0....0....0....0....0....1....0....0....0
..1....1....0....0....0....1....1....0....0....1
..2....2....1....0....0....1....1....1....0....1
..2....2....1....0....2....1....2....2....1....2
..2....2....2....0....2....2....2....2....1....2
MAPLE
k:= 3: N:= 20: # to produce T(n, k) for n=2..N
q:= proc(S, x) local L, m, i;
m:= nops(S);
L:= convert(x+3^m, base, 3)[1..m];
[seq([S[i], L[i]+1], i=1..m)];
end proc:
enlarge:= proc(S) local m, j;
seq(q(S, j), j=0..3^nops(S)-1)
end proc:
States:= map(enlarge, combinat:-powerset([$0..k])): ns:= nops(States):
T:= Matrix(ns, ns):
for j from 1 to ns do
S:= States[j];
if nops(S)=1 and S[1][2]=1 then T[1, j]:= 1 fi
od:
for i from 2 to ns do
S:= States[i]; P:= S[-1];
Sp:= subs(P=[P[1], min(3, P[2]+1)], S);
member(Sp, States, 'j');
T[i, j]:= 1;
for sp from P[1]+1 to k do
Sp:= [op(S), [sp, 1]];
member(Sp, States, 'j');
T[i, j]:= 1
od
od:
v:= Vector[row]([1, 0$(ns-1)]):
good:= proc(s) local L:
L:= map(p -> p[1]$p[2], States[s]);
andmap(j -> member(L[j], [seq(seq(L[i]+L[ip] mod (k+1), ip = {$i+1..nops(L)} minus {j}), i=[$1..j-1, $(j+1)..nops(L)])]),
[$1..nops(L)])
end proc:
goodS:= select(good, [$1..ns]):
vT[0]:= v:
for i from 1 to N+2 do vT[i]:= vT[i-1] . T od:
seq(convert(vT[i][goodS], `+`), i=3..N+2); # Robert Israel, Apr 05 2018
CROSSREFS
Sequence in context: A352833 A034390 A368671 * A144693 A328399 A328171
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin, Jan 07 2011
STATUS
approved

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Last modified May 25 12:29 EDT 2024. Contains 372788 sequences. (Running on oeis4.)