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T(n,k)=Number of length n+3 0..k arrays with no disjoint pairs in any consecutive four terms having the same sum
14

%I #9 Dec 12 2014 20:21:44

%S 8,48,8,168,90,8,440,456,172,8,960,1592,1248,334,8,1848,4344,5796,

%T 3424,656,8,3248,10098,19744,21152,9392,1300,8,5328,20816,55372,89836,

%U 77236,25822,2584,8,8280,39264,133780,303924,408644,282384,71060,5148,8,12320

%N T(n,k)=Number of length n+3 0..k arrays with no disjoint pairs in any consecutive four terms having the same sum

%C Table starts

%C .8....48.....168......440.......960.......1848........3248.........5328

%C .8....90.....456.....1592......4344......10098.......20816........39264

%C .8...172....1248.....5796.....19744......55372......133780.......290004

%C .8...334....3424....21152.....89836.....303924......860360......2143214

%C .8...656....9392....77236....408644....1668072.....5532212.....15837692

%C .8..1300...25822...282384...1859736....9157806....35577396....117045466

%C .8..2584...71060..1032952...8465936...50284864...228817500....865051288

%C .8..5148..195536..3779018..38539276..276119316..1471661464...6393427268

%C .8.10272..537880.13825712.175434372.1516191100..9465023576..47252411120

%C .8.20520.1480026.50587924.798617096.8325624724.60874728614.349232818280

%H R. H. Hardin, <a href="/A247726/b247726.txt">Table of n, a(n) for n = 1..9999</a>

%F Empirical for column k:

%F k=1: a(n) = a(n-1)

%F k=2: a(n) = 2*a(n-1) +2*a(n-4) -4*a(n-5)

%F k=3: [order 27]

%F k=4: [order 45]

%F k=5: [order 76]

%F Empirical for row n:

%F n=1: a(n) = n^4 + 2*n^3 + 3*n^2 + 2*n

%F n=2: a(n) = 4*a(n-1) -4*a(n-2) -4*a(n-3) +10*a(n-4) -4*a(n-5) -4*a(n-6) +4*a(n-7) -a(n-8); also a polynomial of degree 5 plus a linear quasipolynomial with period 2

%F n=3: [order 16; also a polynomial of degree 6 plus a quadratic quasipolynomial with period 12]

%F n=4: [order 34; also a polynomial of degree 7 plus a cubic quasipolynomial with period 420]

%F n=5: [order 72]

%e Some solutions for n=4 k=4

%e ..1....0....3....2....4....2....0....1....1....4....1....0....2....3....3....1

%e ..1....1....0....3....3....1....3....0....0....0....1....3....3....3....0....1

%e ..4....1....1....2....0....4....2....4....4....4....0....0....3....0....3....4

%e ..1....4....1....0....0....0....4....1....4....4....3....4....0....2....2....2

%e ..1....3....1....0....4....0....0....1....4....2....0....2....4....4....4....2

%e ..2....3....0....1....2....2....1....0....0....4....4....1....0....4....0....1

%e ..1....1....1....3....3....3....0....1....2....4....3....2....1....3....0....0

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, Sep 23 2014