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%I A023999 #32 Apr 24 2017 07:11:35
%S A023999 1,5,48,660,11760,257040,6652800,198918720,6745939200,255826771200,
%T A023999 10727081164800,492775291008000,24610605962342400,1327677426915840000,
%U A023999 76940526008586240000,4766815315895592960000,314406967644177408000000,21995911456386651463680000
%N A023999 Absolute value of determinant of n X n matrix whose entries are the integers from 1 to n^2 spiraling inward, starting in a corner.
%C A023999 Starting in the NW or SE corner, the signs are cyclic (+,-,-,+), starting in the NE or SW corner, the signs are always positive.
%H A023999 Alois P. Heinz, <a href="/A023999/b023999.txt">Table of n, a(n) for n = 1..200</a>
%H A023999 Gaurav Bhatnagar, Christian Krattenthaler, <a href="https://arxiv.org/abs/1704.02859">Spiral determinants</a>, arXiv:1704.02859 [math.CO], 2017.
%H A023999 Charles Vanden Eynden, <a href="http://www.jstor.org/stable/2691057">Problem 1517</a>, Mathematics Magazine, Vol. 70, No. 1, Feb., 1997 p. 65.
%F A023999 a(n) = (3n-1) * (2n-3)!/(n-2)! for n >= 2. [corrected by _Robert Israel_, Apr 20 2017]
%F A023999 E.g.f.: ((-2*x-1)*sqrt(1-4*x)+1-4*x)/(16*x-4). - _Robert Israel_, Apr 20 2017
%e A023999 n=4: det of
%e A023999 .1..2..3.4
%e A023999 12.13.14.5
%e A023999 11.16.15.6
%e A023999 10..9..8.7
%p A023999 a:= proc(n) option remember; `if`(n<2, (3*n+1)/4,
%p A023999       4*(3*n-1)*(2*n-5)*(2*n-3) *a(n-2) /(3*n-7))
%p A023999     end:
%p A023999 seq(a(n), n=1..20);  # _Alois P. Heinz_, Jan 21 2014
%t A023999 M[0, 0] = 1;
%t A023999 M[i_, j_] := If[i <= j,
%t A023999   If[i + j >= 0, If[i != j, M[i + 1, j] + 1, M[i, j - 1] + 1],
%t A023999    M[i, j + 1] + 1],
%t A023999   If[i + j > 1, M[i, j - 1] + 1, M[i - 1, j] + 1]
%t A023999   ]
%t A023999 M[n_] := n^2 + 1 - If[EvenQ[n],
%t A023999   Table[M[i, j], {j, n/2, -n/2 + 1, -1}, {i, -n/2 + 1, n/2}],
%t A023999   Table[M[i, j], {j, (n - 1)/2, -(n - 1)/2, -1}, {i, -(n - 1)/2, (n - 1)/2}]]
%t A023999 a[n_]:=Det[M[n]] (* _Christian Krattenthaler_, Apr 19 2017 *)
%o A023999 (Maxima) A023999(n):=if n=1 then 1 else 2*((-1)^((n+4)*(n-1))/2 *(3*n-1) * (2*n-3)!/(n-2)!)$
%o A023999 makelist(A023999(n),n,1,30); /* _Martin Ettl_, Nov 05 2012 */
%Y A023999 Cf. A079340, A078475, A067276, A052182.
%Y A023999 Main diagonal of A226167, A126224 (signed version). - _Alois P. Heinz_, Jan 21 2014
%K A023999 nonn
%O A023999 1,2
%A A023999 Charles Diminnie (charles.diminnie(AT)rampo.angelo.edu)
%E A023999 Edited and extended by _Robert G. Wilson v_, May 07 2003

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