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Number of 1 X n 0..3 arrays avoiding the pattern z-2 z-1 z in any row, column, nw-to-se diagonal or ne-to-sw antidiagonal.
4

%I #22 Jan 29 2025 10:11:43

%S 1,4,16,62,241,936,3636,14124,54865,213124,827884,3215930,12492337,

%T 48526704,188502840,732242616,2844409393,11049158596,42920651992,

%U 166726031798,647650219393,2515808732184,9772703517132,37962239661540,147464991401185,572830367302660

%N Number of 1 X n 0..3 arrays avoiding the pattern z-2 z-1 z in any row, column, nw-to-se diagonal or ne-to-sw antidiagonal.

%C Column and row 1 of A206838.

%H Alois P. Heinz, <a href="/A206839/b206839.txt">Table of n, a(n) for n = 0..1000</a> (terms n = 1..210 from R. H. Hardin)

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,0,-2,1).

%F a(n) = 4*a(n-1) -2*a(n-3) +a(n-4).

%F G.f.: 1 / (1 - 4*x + 2*x^3 - x^4). - _Colin Barker_, Jul 05 2019

%e Some solutions for n=5

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

%p a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <1|-2|0|4>>^n)[4$2]:

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Oct 26 2016

%t LinearRecurrence[{4, 0, -2, 1}, {1, 4, 16, 62}, 30] (* _Jean-François Alcover_, Jan 29 2025 *)

%o (PARI) Vec(1 / (1 - 4*x + 2*x^3 - x^4) + O(x^26)) \\ _Colin Barker_, Jul 05 2019

%Y Cf. A206838.

%K nonn,easy

%O 0,2

%A _R. H. Hardin_, Feb 13 2012

%E a(0)=1 prepended by _Alois P. Heinz_, Oct 26 2016