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A253015
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Sequence of determinants of matrices based on the digits of nonnegative integers.
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1
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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, -1, 1, -1, -5, -11, -19, -29, -41, -55, -71, -4, -1, 4, 1, -4, -11, -20, -31, -44, -59, -9, -5, 1, 9, 5, -1, -9, -19, -31, -45, -16, -11, -4, 5, 16, 11, 4, -5, -16
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OFFSET
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0,3
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COMMENTS
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A given nonnegative integer is transformed into a square matrix whose order equals the quantity of the number's digits. Each element of the main diagonal is a digit of this original number, while other elements are calculated from this diagonal. The determinant of this matrix is the element of the sequence.
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LINKS
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FORMULA
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a(n) = det(B) where B is the n X n matrix with B(i,i) given by the i-th digit of n, B(i,j) = abs(B(i,j-1)-B(i+1,j)) if i < j and B(i,j) = B(i-1,j) + B(i,j+1) if i > j.
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EXAMPLE
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For n=124, a(124)=2, as follows:
B(1,1) = 1;
B(2,2) = 2;
B(3,3) = 4;
B(1,2) = abs(B(1,1) - B(2,2)) = abs(1-2) = 1;
B(2,3) = abs(B(2,2) - B(3,3)) = abs(2-4) = 2;
B(1,3) = abs(B(1,2) - B(2,3)) = abs(1-1) = 1;
B(2,1) = B(1,1) + B(2,2) = 1 + 2 = 3;
B(3,2) = B(2,2) + B(3,3) = 2 + 4 = 6;
B(3,1) = B(2,1) + B(3,2) = 3 + 6 = 9.
Thus,
_______|1 1 1|
B(124)=|3 2 2| --> det(B(124)) = a(124) = 2.
_______|9 6 4|
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CROSSREFS
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See A227876, since the process of matrix construction is this so-called "pyramidalization".
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KEYWORD
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sign,base,easy,dumb
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AUTHOR
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STATUS
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approved
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