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A118713 a(n) = determinant of n X n circulant matrix whose first row is A001358(1), A001358(2), ..., A001358(n) where A001358(n) = n-th semiprime.

%I A118713 #21 Feb 16 2025 08:33:01
%S A118713 4,-20,361,-3567,218053,-3455872,736439027,-16245418225,1519211613654,
%T A118713 -37662452460912,20199655476042865,-643524421698841536,
%U A118713 46513669467992431114,-3754367220494585505280,277686193779526116536293,-123973821931125256333959105,20103033234038999233385180658
%N A118713 a(n) = determinant of n X n circulant matrix whose first row is A001358(1), A001358(2), ..., A001358(n) where A001358(n) = n-th semiprime.
%C A118713 Semiprime analog of A066933 Circulant of prime numbers. a(n) alternates in sign. A048954 Wendt determinant of n-th circulant matrix C(n). A052182 Circulant of natural numbers. A086459 Circulant of powers of 2.
%H A118713 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CirculantMatrix.html">Circulant Matrix</a>.
%e A118713 a(2) = -20 = determinant
%e A118713 |4,6|
%e A118713 |6,4|.
%e A118713 a(3) = 361 = 19^2 = determinant
%e A118713 |4,6,9|
%e A118713 |9,4,6|
%e A118713 |6,9,4|.
%p A118713 A118713 := proc(n)
%p A118713     local C,r,c ;
%p A118713     C := Matrix(1..n,1..n) ;
%p A118713     for r from 1 to n do
%p A118713     for c from 1 to n do
%p A118713         C[r,c] := A001358(1+((c-r) mod n)) ;
%p A118713     end do:
%p A118713     end do:
%p A118713     LinearAlgebra[Determinant](C) ;
%p A118713 end proc:
%p A118713 seq(A118713(n),n=1..13) ;
%t A118713 nmax = 13;
%t A118713 sp = Select[Range[3 nmax], PrimeOmega[#] == 2&];
%t A118713 a[n_] := Module[{M}, M[1] = sp[[1 ;; n]];
%t A118713    M[k_] := M[k] = RotateRight[M[k - 1]];
%t A118713    Det[Table[M[k], {k, 1, n}]]];
%t A118713 Table[a[n], {n, 1, nmax}] (* _Jean-François Alcover_, Feb 16 2023 *)
%Y A118713 Cf. A001358, A048954, A052182, A066933, A086459, A086569.
%K A118713 easy,sign
%O A118713 1,1
%A A118713 _Jonathan Vos Post_, May 20 2006
%E A118713 Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Aug 23 2007