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%I #4 Nov 30 2014 18:09:46
%S 5832,34992,209952,1259712,4665600,17280000,64000000,180000000,
%T 506250000,1423828125,3348843750,7876480500,18525482136,38423222208,
%U 79692609024,165288374272,312264916992,589934886912,1114512556032,1959104102400
%N Number of (n+2)X(3+2) 0..1 arrays with nondecreasing sum of every three consecutive values in every row and column
%C Column 3 of A251192
%H R. H. Hardin, <a href="/A251189/b251189.txt">Table of n, a(n) for n = 1..210</a>
%F Empirical: a(n) = 2*a(n-1) -a(n-2) +14*a(n-3) -28*a(n-4) +14*a(n-5) -91*a(n-6) +182*a(n-7) -91*a(n-8) +364*a(n-9) -728*a(n-10) +364*a(n-11) -1001*a(n-12) +2002*a(n-13) -1001*a(n-14) +2002*a(n-15) -4004*a(n-16) +2002*a(n-17) -3003*a(n-18) +6006*a(n-19) -3003*a(n-20) +3432*a(n-21) -6864*a(n-22) +3432*a(n-23) -3003*a(n-24) +6006*a(n-25) -3003*a(n-26) +2002*a(n-27) -4004*a(n-28) +2002*a(n-29) -1001*a(n-30) +2002*a(n-31) -1001*a(n-32) +364*a(n-33) -728*a(n-34) +364*a(n-35) -91*a(n-36) +182*a(n-37) -91*a(n-38) +14*a(n-39) -28*a(n-40) +14*a(n-41) -a(n-42) +2*a(n-43) -a(n-44)
%F Empirical for n mod 3 = 0: a(n) = (1/918330048)*n^15 + (31/306110016)*n^14 + (445/102036672)*n^13 + (3923/34012224)*n^12 + (23747/11337408)*n^11 + (104525/3779136)*n^10 + (345503/1259712)*n^9 + (873041/419904)*n^8 + (26558/2187)*n^7 + (159289/2916)*n^6 + (91187/486)*n^5 + (78251/162)*n^4 + (8116/9)*n^3 + 1152*n^2 + 900*n + 324
%F Empirical for n mod 3 = 1: a(n) = (1/918330048)*n^15 + (31/306110016)*n^14 + (335/76527504)*n^13 + (26785/229582512)*n^12 + (328165/153055008)*n^11 + (4405193/153055008)*n^10 + (66934501/229582512)*n^9 + (57891005/25509168)*n^8 + (1396581025/102036672)*n^7 + (58738095125/918330048)*n^6 + (4395145625/19131876)*n^5 + (2978865625/4782969)*n^4 + (17703500000/14348907)*n^3 + (8065000000/4782969)*n^2 + (6800000000/4782969)*n + (8000000000/14348907)
%F Empirical for n mod 3 = 2: a(n) = (1/918330048)*n^15 + (31/306110016)*n^14 + (445/102036672)*n^13 + (105941/918330048)*n^12 + (641665/306110016)*n^11 + (942593/34012224)*n^10 + (252901303/918330048)*n^9 + (213715055/102036672)*n^8 + (6532447/531441)*n^7 + (12779491795/229582512)*n^6 + (1845484403/9565938)*n^5 + (801235309/1594323)*n^4 + (13667452400/14348907)*n^3 + (5931391984/4782969)*n^2 + (527067520/531441)*n + (5270675200/14348907)
%e Some solutions for n=2
%e ..0..0..1..1..1....0..0..0..0..0....0..0..1..1..0....0..0..0..0..0
%e ..0..0..0..1..0....1..0..0..1..1....0..1..1..1..1....0..1..0..0..1
%e ..0..0..0..1..0....1..0..0..1..0....0..0..0..1..1....0..0..0..1..1
%e ..1..1..1..1..1....0..1..0..0..1....0..1..1..1..1....0..1..0..1..1
%K nonn
%O 1,1
%A _R. H. Hardin_, Nov 30 2014
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