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A181208 Number of n X 4 binary matrices with no two 1's adjacent diagonally or antidiagonally.

%I A181208 #17 Aug 29 2024 17:08:30
%S A181208 16,64,484,2704,17424,104976,652864,4000000,24681024,151782400,
%T A181208 934891776,5754132736,35428274176,218096472064,1342706197504,
%U A181208 8266039005184,50888705511424,313286601609216,1928696564957184,11873676328960000
%N A181208 Number of n X 4 binary matrices with no two 1's adjacent diagonally or antidiagonally.
%C A181208 Column 4 of A181212.
%H A181208 Robert Israel, <a href="/A181208/b181208.txt">Table of n, a(n) for n = 1..1265</a> (n = 1..325 from R. H. Hardin)
%H A181208 Robert Israel, <a href="/A181208/a181208.pdf">Maple-assisted proof of formula</a>
%H A181208 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,8,-48,24,32,-16).
%F A181208 Empirical: a(n) = 6*a(n-1) + 8*a(n-2) - 48*a(n-3) + 24*a(n-4) + 32*a(n-5) - 16*a(n-6).
%F A181208 Formula confirmed by _Robert Israel_, Dec 25 2017 (see link).
%F A181208 G.f.: 4*x*(4 - 8*x - 7*x^2 + 14*x^3 + 4*x^4 - 4*x^5) / ((1 - 8*x + 12*x^2 - 4*x^3)*(1 + 2*x - 4*x^2 - 4*x^3)). - _Colin Barker_, Mar 26 2018
%p A181208 f:= gfun:-rectoproc({a(n)=6*a(n-1)+8*a(n-2)-48*a(n-3)+24*a(n-4)+32*a(n-5)-16*a(n-6), a(1)=16, a(2)=64, a(3)=484, a(4)=2704, a(5)=17424, a(6)=104976},a(n),remember):
%p A181208 map(f, [$1..20]); # _Robert Israel_, Dec 25 2017
%t A181208 RecurrenceTable[{a[n] == 6*a[n-1] + 8*a[n-2] - 48*a[n-3] + 24*a[n-4] + 32*a[n-5] - 16*a[n-6], a[1] == 16, a[2] == 64, a[3] == 484, a[4] == 2704, a[5] == 17424, a[6] == 104976}, a, {n, 1, 20}] (* _Jean-François Alcover_, Aug 29 2022, after _Robert Israel_ *)
%t A181208 LinearRecurrence[{6,8,-48,24,32,-16},{16,64,484,2704,17424,104976},30] (* _Harvey P. Dale_, Aug 29 2024 *)
%o A181208 (PARI) Vec(4*x*(4 - 8*x - 7*x^2 + 14*x^3 + 4*x^4 - 4*x^5) / ((1 - 8*x + 12*x^2 - 4*x^3)*(1 + 2*x - 4*x^2 - 4*x^3)) + O(x^30)) \\ _Colin Barker_, Mar 26 2018
%Y A181208 Cf. A181212.
%K A181208 nonn,easy
%O A181208 1,1
%A A181208 _R. H. Hardin_, Oct 10 2010