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%I #6 Sep 18 2014 14:20:34
%S 8,33,8,88,45,8,185,136,61,8,336,317,220,81,8,553,600,561,364,105,8,
%T 848,1033,1124,1007,604,153,8,1233,1616,2009,2164,1823,1018,217,8,
%U 1720,2409,3220,3997,4228,3455,1732,297,8,2321,3400,4901,6584,8051,8440,6495,2956
%N T(n,k)=Number of length n+3 0..k arrays with some disjoint pairs in every consecutive four terms having the same sum
%C Table starts
%C .8..33...88...185....336....553....848....1233....1720....2321....3048....3913
%C .8..45..136...317....600...1033...1616....2409....3400....4661....6168....8005
%C .8..61..220...561...1124...2009...3220....4901....7016....9737...13000...17025
%C .8..81..364..1007...2164...3997...6584...10219...14852...20847...28108...37095
%C .8.105..604..1823...4228...8051..13668...21609...31924...45309...61740...82067
%C .8.153.1018..3455...8440..16683..29012...47061...70374..101211..139098..186709
%C .8.217.1732..6495..16932..34695..62108..103013..156308..227701..316236..428111
%C .8.297.2956.12105..34068..72269.133716..226309..349160..515043..723892..987667
%C .8.393.5050.22459..68688.150677.288996..498569..783568.1170169.1665908.2290065
%C .8.585.8638.43255.139040.318575.627654.1111891.1772920.2686215.3862654.5366083
%H R. H. Hardin, <a href="/A247533/b247533.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) = a(n-1) +4*a(n-4) -4*a(n-5)
%F k=3: a(n) = 2*a(n-1) -a(n-3) +a(n-4) -a(n-5) -a(n-6) +a(n-7)
%F k=4: [order 29] for n>30
%F k=5: [order 56]
%F k=6: [order 82] for n>84
%F Empirical for row n:
%F n=1: a(n) = 2*n^3 + 3*n^2 + 2*n + 1
%F n=2: a(n) = 2*a(n-1) +a(n-2) -4*a(n-3) +a(n-4) +2*a(n-5) -a(n-6); also a polynomial of degree 3 plus a linear quasipolynomial with period 2
%F n=3: [recurrence of order 12; also a polynomial of degree 3 plus a linear quasipolynomial with period 12]
%F n=4: [recurrence of order 24; also a polynomial of degree 3 plus a linear quasipolynomial with period 420]
%F n=5: [recurrence of order 48; also a polynomial of degree 3 plus a linear quasipolynomial with period 27720; note 2 12 420 27720 matches A060942]
%F n=6: [recurrence of order 92]
%e Some solutions for n=6 k=4
%e ..2....3....2....1....4....3....3....0....2....1....0....3....1....4....1....1
%e ..1....2....1....2....1....2....2....1....4....1....0....0....3....2....2....4
%e ..0....3....3....2....0....2....2....2....3....2....1....2....4....1....1....1
%e ..3....2....2....1....3....1....3....3....1....2....1....1....2....3....2....4
%e ..4....3....2....3....2....1....3....0....2....3....2....1....3....2....3....1
%e ..1....2....3....2....1....2....4....1....2....1....0....2....3....4....4....4
%e ..2....3....3....2....2....2....2....2....3....0....1....0....4....3....3....1
%e ..3....4....4....1....3....3....3....1....1....2....3....1....4....3....2....4
%e ..4....1....2....3....4....3....1....0....2....3....4....1....3....4....1....1
%Y Row 1 is A212133(n+1)
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_, Sep 18 2014
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