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%I #47 Apr 24 2024 12:57:43
%S 1,-5,78,-1200,19680,-351360,6854400,-145797120,3367526400,
%T -84072038400,2258332876800,-64990937088000,1995834890649600,
%U -65167516237824000,2254974602969088000,-82443156980760576000,3176032637949050880000,-128603097714237898752000,5460911310769351557120000
%N Determinant of the n X n matrix formed by placing 1..n^2 in L-shaped gnomons in alternating directions.
%C The matrix is the upper-left n X n part of the square arrangement in A081344.
%C Number i is in the matrix at row A220604(i) column A220603(i), for i = 1..n^2.
%C Conjecture: a(n) has trailing zeros for n > 3. - _Stefano Spezia_, May 31 2023
%C The conjecture is true and its proof follows easily from _Detlef Meya_'s formula. - _Stefano Spezia_, Apr 20 2024
%H Stefano Spezia, <a href="/A363376/b363376.txt">Table of n, a(n) for n = 1..400</a>
%H Nicolay Avilov, <a href="/A363376/a363376_1.jpg">Illustration of a(1)-a(5)</a>
%F a(1) = 1, for a > 1: a(n) = (-1)^(n-1)*2^(n-3)*(2*n*(n-1)+1)*(n!). - _Detlef Meya_, Jun 11 2023
%F E.g.f.: x*(2 + 7*x + 20*x^2 + 12*x^3)/(2*(1 + 2*x)^3). - _Stefano Spezia_, Apr 20 2024
%e | 1----2 9---10 25 |
%e | | | | | |
%e | 4----3 8 11 24 |
%e | | | | | |
%e a(5) = | 5----6----7 12 23 | = 19680.
%e | | | |
%e | 16---15---14---13 22 |
%e | | | |
%e | 17---18---19---20---21 |
%t a={}; For[n=1, n<=19, n++,k=i=j=1; M[i,j]=k++; For[h=1, h<n, h++, If[i==j==1, M[i,++j]=k++;For[c=1, c<=h, c++, M[++i,j]=k++; M[i, --j]=k++], If[j==1 && i!=1, M[++i, j]=k++; For[c=1, c<=h, c++, M[i, ++j]=k++];For[c=1, c<=h, c++, M[--i, j]=k++], If[i==1 &&j!=1, M[i, ++j]=k++; For[c=1, c<=h, c++, M[++i, j]=k++]; For[c=1, c<=h, c++, M[i, --j]=k++]]]]];AppendTo[a, Det[Table[M[i, j], {i, n}, {j, n}]]]]; a (* _Stefano Spezia_, May 31 2023 *)
%t a={1};For[n=2,n<20,n++,AppendTo[a,(-1)^(n-1)*2^(n-3)*(2*n*(n-1)+1)*n!]];a (* _Detlef Meya_, Jun 11 2023 *)
%Y Cf. A081344, A220603, A220604, A363460 (permanent).
%K sign
%O 1,2
%A _Nicolay Avilov_, May 29 2023
%E a(16)-a(19) from _Stefano Spezia_, May 31 2023
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