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A268457 T(n,k)=Number of length-n 0..k arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.

%I A268457 #4 Feb 04 2016 22:05:30
%S A268457 2,3,4,4,9,7,5,16,25,11,6,25,61,67,16,7,36,121,229,176,22,8,49,211,
%T A268457 581,852,456,29,9,64,337,1231,2776,3146,1169,37,10,81,505,2311,7160,
%U A268457 13204,11536,2971,46,11,100,721,3977,15816,41526,62535,42032,7496,56,12,121
%N A268457 T(n,k)=Number of length-n 0..k arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.
%C A268457 Table starts
%C A268457 ..2.....3......4.......5........6.........7.........8..........9.........10
%C A268457 ..4.....9.....16......25.......36........49........64.........81........100
%C A268457 ..7....25.....61.....121......211.......337.......505........721........991
%C A268457 .11....67....229.....581.....1231......2311......3977.......6409.......9811
%C A268457 .16...176....852....2776.....7160.....15816.....31276......56912......97056
%C A268457 .22...456...3146...13204....41526....108032....245626.....504876.....959414
%C A268457 .29..1169..11536...62535...240170....736525...1926444....4474451....9476950
%C A268457 .37..2971..42032..294967..1385338...5012171..15089356...39616567...93543782
%C A268457 .46..7496.152254.1385969..7970326..34047931.118040270..350431909..922677334
%C A268457 .56.18796.548568.6488635.45742764.230889543.922247248.3096903363.9094484100
%H A268457 R. H. Hardin, <a href="/A268457/b268457.txt">Table of n, a(n) for n = 1..421</a>
%F A268457 Empirical for column k:
%F A268457 k=1: a(n) = (1/2)*n^2 + (1/2)*n + 1
%F A268457 k=2: a(n) = 6*a(n-1) -13*a(n-2) +14*a(n-3) -10*a(n-4) +4*a(n-5) -a(n-6)
%F A268457 k=3: [order 15]
%F A268457 k=4: [order 28]
%F A268457 k=5: [order 51]
%F A268457 k=6: [order 89]
%F A268457 Empirical for row n:
%F A268457 n=1: a(n) = n + 1
%F A268457 n=2: a(n) = n^2 + 2*n + 1
%F A268457 n=3: a(n) = n^3 + 3*n^2 + 2*n + 1
%F A268457 n=4: a(n) = n^4 + 4*n^3 + 4*n^2 + n + 1
%F A268457 n=5: a(n) = n^5 + 5*n^4 + 7*n^3 + n^2 + 2*n
%F A268457 n=6: a(n) = n^6 + 6*n^5 + 11*n^4 + 2*n^3 + 6*n - 4
%F A268457 n=7: a(n) = n^7 + 7*n^6 + 16*n^5 + 5*n^4 - 7*n^3 + 18*n^2 - 5*n - 5 for n>1
%e A268457 Some solutions for n=6 k=4
%e A268457 ..3....2....3....2....2....0....4....2....2....0....1....0....0....3....4....3
%e A268457 ..0....2....1....2....4....0....1....1....2....4....4....2....3....2....3....3
%e A268457 ..0....4....1....1....4....2....3....0....1....4....2....1....1....2....0....1
%e A268457 ..3....0....4....2....3....2....1....0....2....4....4....4....3....1....0....0
%e A268457 ..4....1....2....3....2....2....0....4....0....4....4....4....1....3....1....3
%e A268457 ..0....1....3....4....3....1....4....0....2....0....1....1....1....2....4....2
%Y A268457 Column 1 is A000124.
%Y A268457 Row 1 is A000027(n+1).
%Y A268457 Row 2 is A000290(n+1).
%Y A268457 Row 3 is A061600(n+1).
%K A268457 nonn,tabl
%O A268457 1,1
%A A268457 _R. H. Hardin_, Feb 04 2016