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A246479 T(n,k)=Number of length n+3 0..k arrays with no pair in any consecutive four terms totalling exactly k.

%I A246479 #10 Jul 23 2025 11:31:20
%S A246479 2,10,2,60,14,2,172,132,20,2,462,484,292,28,2,966,1734,1376,644,38,2,
%T A246479 1880,4386,6534,3904,1420,52,2,3256,10376,20004,24582,11020,3132,72,2,
%U A246479 5370,20840,57416,91212,92478,31104,6908,100,2,8290,39690,133664,317576
%N A246479 T(n,k)=Number of length n+3 0..k arrays with no pair in any consecutive four terms totalling exactly k.
%C A246479 Table starts
%C A246479 .2..10....60.....172......462.......966.......1880........3256.........5370
%C A246479 .2..14...132.....484.....1734......4386......10376.......20840........39690
%C A246479 .2..20...292....1376.....6534.....20004......57416......133664.......293770
%C A246479 .2..28...644....3904....24582.....91212.....317576......857248......2174090
%C A246479 .2..38..1420...11020....92478....415650....1756472.....5497304.....16089370
%C A246479 .2..52..3132...31104...347934...1893780....9714968....35251360....119069850
%C A246479 .2..72..6908...87888..1309038...8628792...53733080...226048032....881180090
%C A246479 .2.100.15236..248568..4924998..39320988..297195272..1449551536...6521200010
%C A246479 .2.138.33604..702724.18529350.179184654.1643773832..9295405128..48260338570
%C A246479 .2.190.74116.1985932.69713094.816514170.9091640072.59607621016.357152100490
%H A246479 R. H. Hardin, <a href="/A246479/b246479.txt">Table of n, a(n) for n = 1..9999</a>
%F A246479 Empirical for column k:
%F A246479 k=1: a(n) = a(n-1)
%F A246479 k=2: a(n) = a(n-1) +a(n-4)
%F A246479 k=3: a(n) = 2*a(n-1) +a(n-3)
%F A246479 k=4: a(n) = 2*a(n-1) +a(n-3) +14*a(n-4) +3*a(n-5) +6*a(n-6) +a(n-8) +a(n-9)
%F A246479 k=5: a(n) = 3*a(n-1) +2*a(n-2) +3*a(n-3) +a(n-4)
%F A246479 k=6: [order 10]
%F A246479 k=7: a(n) = 5*a(n-1) +2*a(n-2) +5*a(n-3) +a(n-4)
%F A246479 k=8: [order 10]
%F A246479 k=9: a(n) = 7*a(n-1) +2*a(n-2) +7*a(n-3) +a(n-4)
%F A246479 From _Robert Israel_, Nov 10 2024: (Start)
%F A246479 It appears that for k >= 5 odd, the recurrence for column k is
%F A246479 a(n) = (k - 2)*a(n-1) + 2*a(n-2) + (k - 2)*a(n-3) + a(n-4)
%F A246479 and that for k >= 6 even, the recurrence for column k is
%F A246479 a(n) = (k - 3)*a(n-1) + 2*a(n-2) + (k-3)*a(n-3) + (k^3 - 6*k^2 + 15*k - 13)*a(n-4) + (3*k^2 - 11*k + 13)*a(n-5) + (k^3 - 7*k^2 + 19*k - 19)*a(n-6) + (k^2 - 4*k + 6)*a(n-7) + a(n-8) + (k - 2)*a(n-9) + a(n-10). (End)
%F A246479 Empirical for row n:
%F A246479 n=1: a(n) = 3*a(n-1) -a(n-2) -5*a(n-3) +5*a(n-4) +a(n-5) -3*a(n-6) +a(n-7)
%F A246479 n=2: a(n) = 3*a(n-1) -8*a(n-3) +6*a(n-4) +6*a(n-5) -8*a(n-6) +3*a(n-8) -a(n-9)
%F A246479 n=3: [order 11]
%F A246479 n=4: [order 13]
%F A246479 n=5: [order 15]
%F A246479 n=6: [order 17]
%F A246479 n=7: [order 19]
%e A246479 Some solutions for n=5 k=4
%e A246479 ..0....3....0....2....2....2....3....2....2....0....4....4....0....0....1....0
%e A246479 ..2....4....0....3....0....4....4....1....0....1....3....3....0....0....0....2
%e A246479 ..1....3....2....0....3....4....2....0....0....0....3....3....0....3....2....1
%e A246479 ..0....4....0....3....3....3....4....0....0....1....3....4....1....2....1....1
%e A246479 ..0....4....1....0....3....4....4....0....1....1....3....3....0....3....1....4
%e A246479 ..0....3....0....0....4....3....4....1....0....0....3....4....1....4....4....4
%e A246479 ..1....3....1....2....3....3....4....0....0....1....2....4....0....3....1....1
%e A246479 ..0....3....1....3....3....4....4....1....1....2....3....3....1....4....1....1
%p A246479 G:= proc(m,k) # first m terms in column k
%p A246479   local q,r,s,S,nS,M,u,v,V,i;
%p A246479   S:= remove(t -> t[1]+t[2]=k or t[1]+t[3]=k or t[2]+t[3]=k, [seq(seq(seq([q,r,s],s=0..k),r=0..k),q=0..k)]);
%p A246479 nS:= nops(S);
%p A246479 M:= Matrix(nS,nS,(i,j) -> `if`(S[i][2..3] = S[j][1..2] and S[i][1] + S[j][3] <> k, 1, 0));
%p A246479   u:= Vector[column](nS,1); v:= u;
%p A246479   V:= Vector(m);
%p A246479   for i from 1 to m  do
%p A246479     v:= M . v;
%p A246479     V[i]:= u^%T . v
%p A246479   od;
%p A246479   V
%p A246479 end proc:
%p A246479 R:= Matrix(10,20):
%p A246479 interface(rtablesize=[10,20]):
%p A246479 for j from 1 to 20 do R[.., j] := G(10, j) od:
%p A246479 R; # _Robert Israel_, Nov 10 2024
%K A246479 nonn,tabl
%O A246479 1,1
%A A246479 _R. H. Hardin_, Aug 27 2014