A220642 - OEIS

%I #11 Nov 03 2023 15:43:20

%S 1,13,3096,373177,53841725,7444342896,1040315976961,145000880411157,

%T 20223491612180232,2820152941289640505,393283923444213896309,

%U 54844809649495130675968,7648317475647716579501281,1066586359952790876210231837,148739462164292054050115639320

%N Number of ways to reciprocally link elements of an n X 6 array either to themselves or to exactly one king-move neighbor.

%C Column 6 of A220644.

%H Alois P. Heinz, <a href="/A220642/b220642.txt">Table of n, a(n) for n = 0..300</a> (terms n = 1..210 from R. H. Hardin)

%F Empirical: a(n) = 114*a(n-1) +4256*a(n-2) -91572*a(n-3) -1178554*a(n-4) +30456760*a(n-5) -130975946*a(n-6) -821286442*a(n-7) +8116096176*a(n-8) -10363446884*a(n-9) -106454109710*a(n-10) +356958898120*a(n-11) +328038015993*a(n-12) -2687705043916*a(n-13) +1546010691232*a(n-14) +7214129736088*a(n-15) -10475377691972*a(n-16) -2990479177712*a(n-17) +18333779754964*a(n-18) -16202802907924*a(n-19) -5141934133696*a(n-20) +21799390094104*a(n-21) -11778527057052*a(n-22) -4558878158992*a(n-23) +6009171204129*a(n-24) -2343244613126*a(n-25) +703880375232*a(n-26) +353889755932*a(n-27) -387967483426*a(n-28) +53576735032*a(n-29) -17639703466*a(n-30) -2578302306*a(n-31) -976631280*a(n-32) +921279564*a(n-33) -56415798*a(n-34) -4701240*a(n-35) -552825*a(n-36).

%F G.f.: see Maple program. - _Alois P. Heinz_, Jun 03 2014

%e Some solutions for n=3 0=self 1=nw 2=n 3=ne 4=w 6=e 7=sw 8=s 9=se (reciprocal directions total 10)

%e ..0..7..8..6..4..8....0..7..7..0..0..8....0..9..6..4..7..0....0..0..7..9..0..0

%e ..3..9..2..0..0..2....3..3..0..6..4..2....8..9..1..3..0..8....0..3..0..0..1..8

%e ..0..0..1..0..0..0....6..4..0..6..4..0....2..0..1..0..0..2....0..0..0..0..0..2

%p gf:= -(42525*x^34 +364905*x^33 +4427406*x^32 -69988761*x^31 +75088869*x^30 +126251376*x^29 +1409947907*x^28 -3807220353*x^27 +31562787626*x^26 -34451027911*x^25 -29205077493*x^24 +161219121840*x^23 -514135270654*x^22 +487268729962*x^21 +681687943708*x^20 -1511580215802*x^19 +660828588610*x^18

%p +669167562768*x^17 -1110589682746*x^16 +414093064814*x^15 +401344851300*x^14 -357570201838*x^13 -8972200506*x^12 +82109485328*x^11 -17558268975*x^10 -5482245411*x^9 +2504769654*x^8 -169204765*x^7 -66910711*x^6 +13483712*x^5 -491961*x^4 -56477*x^3 +2642*x^2 +101*x -1) / (552825*x^36 +4701240*x^35 +56415798*x^34 -921279564*x^33 +976631280*x^32 +2578302306*x^31

%p +17639703466*x^30 -53576735032*x^29 +387967483426*x^28 -353889755932*x^27 -703880375232*x^26 +2343244613126*x^25 -6009171204129*x^24 +4558878158992*x^23 +11778527057052*x^22 -21799390094104*x^21 +5141934133696*x^20 +16202802907924*x^19 -18333779754964*x^18

%p +2990479177712*x^17 +10475377691972*x^16 -7214129736088*x^15 -1546010691232*x^14 +2687705043916*x^13 -328038015993*x^12 -356958898120*x^11 +106454109710*x^10 +10363446884*x^9 -8116096176*x^8 +821286442*x^7 +130975946*x^6 -30456760*x^5 +1178554*x^4 +91572*x^3 -4256*x^2 -114*x +1):

%p a:= n-> coeff(series(gf, x, n+1), x, n):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Jun 03 2014

%K nonn

%O 0,2

%A _R. H. Hardin_, Dec 17 2012