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A177894 Determinant of the square matrix whose rows are the cyclic permutations of the digits of n.

%I A177894 #25 Mar 11 2024 04:13:00
%S A177894 0,1,2,3,4,5,6,7,8,9,1,0,-3,-8,-15,-24,-35,-48,-63,-80,4,3,0,-5,-12,
%T A177894 -21,-32,-45,-60,-77,9,8,5,0,-7,-16,-27,-40,-55,-72,16,15,12,7,0,-9,
%U A177894 -20,-33,-48,-65,25,24,21,16,9,0,-11,-24,-39,-56,36,35,32,27,20,11,0,-13,-28,-45,49,48,45,40,33,24,13,0,-15,-32,64,63,60,55,48,39,28,15,0,-17,81,80,77,72,65,56,45,32
%N A177894 Determinant of the square matrix whose rows are the cyclic permutations of the digits of n.
%H A177894 David A. Corneth, <a href="/A177894/b177894.txt">Table of n, a(n) for n = 0..9999</a>
%F A177894 For n = a, det(M) = a;
%F A177894 for n = ab, det(M) = a^2 - b^2;
%F A177894 for n = abc, det(M) = 3abc - a^3 - b^3 - c^3; ...
%e A177894 for n=104, the (3 X 3) matrix M is
%e A177894   [1 0 4]
%e A177894   [0 4 1]
%e A177894   [4 1 0]
%e A177894 and a(104) = det(M) = -65.
%t A177894 A177894[n_] := If[n < 10, n, Det[NestList[RotateLeft, IntegerDigits[n], IntegerLength[n]-1]]]; Array[A177894, 100, 0] (* _Paolo Xausa_, Mar 11 2024 *)
%o A177894 (Sage)
%o A177894 def A177894(n):
%o A177894     d = n.digits()[::-1] if n > 0 else [0]
%o A177894     M = Matrix(lambda i,j: d[(i+j) % len(d)], nrows=len(d))
%o A177894     return M.determinant() # _D. S. McNeil_, Dec 16 2010
%o A177894 (PARI) a(n) = {if(n<10, return(n)); my(d = digits(n), m, s); d = concat(d, d); s = #d/2; m = matrix(s, s, i, j, d[i+j-1]); matdet(m)} \\ _David A. Corneth_, Jun 12 2017
%Y A177894 Coincides with A257587 for the first 100 terms, but differs thereafter.
%K A177894 sign,base,easy,look
%O A177894 0,3
%A A177894 _Michel Lagneau_, Dec 15 2010