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.
1, 5, 48, 660, 11760, 257040, 6652800, 198918720, 6745939200, 255826771200, 10727081164800, 492775291008000, 24610605962342400, 1327677426915840000, 76940526008586240000, 4766815315895592960000, 314406967644177408000000, 21995911456386651463680000
Offset: 1
Keywords
Examples
n=4: det of .1..2..3.4 12.13.14.5 11.16.15.6 10..9..8.7
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..200
- Gaurav Bhatnagar, Christian Krattenthaler, Spiral determinants, arXiv:1704.02859 [math.CO], 2017.
- Charles Vanden Eynden, Problem 1517, Mathematics Magazine, Vol. 70, No. 1, Feb., 1997 p. 65.
Crossrefs
Programs
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Maple
a:= proc(n) option remember; `if`(n<2, (3*n+1)/4, 4*(3*n-1)*(2*n-5)*(2*n-3) *a(n-2) /(3*n-7)) end: seq(a(n), n=1..20); # Alois P. Heinz, Jan 21 2014 -
Mathematica
M[0, 0] = 1; M[i_, j_] := If[i <= j, If[i + j >= 0, If[i != j, M[i + 1, j] + 1, M[i, j - 1] + 1], M[i, j + 1] + 1], If[i + j > 1, M[i, j - 1] + 1, M[i - 1, j] + 1] ] M[n_] := n^2 + 1 - If[EvenQ[n], Table[M[i, j], {j, n/2, -n/2 + 1, -1}, {i, -n/2 + 1, n/2}], Table[M[i, j], {j, (n - 1)/2, -(n - 1)/2, -1}, {i, -(n - 1)/2, (n - 1)/2}]] a[n_]:=Det[M[n]] (* Christian Krattenthaler, Apr 19 2017 *) -
Maxima
A023999(n):=if n=1 then 1 else 2*((-1)^((n+4)*(n-1))/2 *(3*n-1) * (2*n-3)!/(n-2)!)$ makelist(A023999(n),n,1,30); /* Martin Ettl, Nov 05 2012 */
Formula
a(n) = (3n-1) * (2n-3)!/(n-2)! for n >= 2. [corrected by Robert Israel, Apr 20 2017]
E.g.f.: ((-2*x-1)*sqrt(1-4*x)+1-4*x)/(16*x-4). - Robert Israel, Apr 20 2017
Extensions
Edited and extended by Robert G. Wilson v, May 07 2003
Comments