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 A164516 Infinite set of Petoukhov 2^n x 2^n Petoukhov matrices by antidiagonals, generated from w = (-.5 + sqrt(-3)/2) 1
 -1, 2, 2, -1, 1, -2, -2, 4, 1, 4, -2, -2, -2, -2, 4, 1, 4, -2, -2, 1, -1, 2, 2, -4, -1, -4, 2, 2, 2, 2, -4, -4, -1, -4, -4, 8, 8, 2, 2, 8, 8, -4, -4, -4, -1, -4, -4, -4, 2, 2, 2, 2, 2, 2, 2, 2, -4, -4, -4, -1, -4, -4, -4, -4, 8, 8, 2, 2, 8, 8, -4, -4, -1, -4, -4, 2, 2, 2, 2, -4, -1, -4, 2, 2 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Sergei Petoukhov has pioneered the investigation of certain matrices whose square roots are irrational numbers; and in recognition his discoveries such matrices and their accompanying sequences may be termed "Petoukhov matrices/sequences". Refer to A119633 for a related sequence. REFERENCES Sergey Petoukhov & Matthew He, "symmetrical Analysis Techniques for Genetics systems and Bioinformatics, Advanced Patterns & Applications", IGI Global, 978-1-60566-127-9, October, 2009, Chapters 2, 4, and 6. LINKS FORMULA Given w = (-.5 + sqrt(-3)/2), use the exponent codes of A164092 to create alternating circulant matrices such that a row with 2^n terms generates 2^n x 2^n matrices. Terms in these matrices = exponents for w, then square the matrices. Sequence A164516 = antidiagonals of the infinite set of 2^n x 2^n matrices, exhausting terms in the n-th matrix before using the terms of the next matrix. EXAMPLE The exponent codes of A164092 are: . 0; (skip as trivial); 1, -1; (creates the 2x2 matrix [w,1/w; 1/w,w](exponents of w = 1 & -1). 2, 0, -2, 0; 3, 1, -1, 1, -1, -3, -1, 1; 4, 3, .0, 2, .0, -2, .0, 2, 0, -2, -4, -2, 0, -2, 0, 2; ... Exponent codes (above) are generated by adding "1" to each term in n-th row bringing down that subset as the first half of the next row. Second half of the next (n+1)-th) row is created by reversing the terms of n-th row and subtracting "1" from each term. (2, 0, -2, 0) becomes (3, 1, -1, 1) as the first half of the next row. Then append (-1, -3, -1, 1), getting (3, 1, -1, 1, -1, -3, -1, 1) as row 3. Let these rows = "A" for each matrix . In a 2^n * 2^n matrix with a conventional upper left term of (1,1), place A as the top row and left column. Put leftmost term of A into every (n,n) (i.e. diagonal position). Then, odd columns are circulated from position (n,n) downwards while even columns circulate upwards starting from (n,n). Using A with 8 terms we obtain the following 8x8 matrix: . 3, 1, -1, 1, -1, -3, -1, 1; 1, 3, 1, -1, -3, -1, 1, -1; -1, 1, 3, 1, -1, 1, -1, -3; 1, -1, 1, 3, 1, -1, -3, -1; -1, -3, -1, 1, 3, 1, -1, 1; -3, -1, 1, -1, 1, 3, 1, -1; -1, 1, -1, -3, -1, 1, 3, 1; 1, -1, -3, -1, 1, -1, 1, 3; . The foregoing terms are exponents to w, so our new matrix becomes: . 1, w, 1/w, w, 1/w, 1, 1/w, w; w, 1, w, 1/w, 1, 1/w, w, 1/w; 1/w, w, 1, w, 1/w, w, 1/w, 1; w, 1/w, w, 1, w, 1/w, 1, 1/w; 1/w, 1, 1/w, w, 1, w, 1/w, w; 1, 1/w, w, 1/w, w, 1, w, 1/w; 1/w, w, 1/w, 1, 1/w, w, 1, w; w, 1/w, 1, 1/w, w, 1/w, w, 1; . Let the foregoing matrix = Q, then take Q^2 = . -1, 2, -4, 2, -4, 8, -4, 2; 2, -1, 2, -4, 8, -4, 2, -4; -4, 2, -1, 2, -4, 2, -4, 8; 2, -4, 2, -1, 2, -4, 8, -4; -4, 8, -4, 2, -1, 2, -4, 2; 8, -4, 2, -4, 2, -1, 2, -4; -4, 2, -4, 8, -4, 2, -1, 2; 2, -4, 8, -4, 2, -4, 2, -1; . Following analogous procedures for the 2x2 and 4x4 matrices, those are [ -1, 2; 2,-1], and . 1, -2, 4, -2; -2, 1, -2, 4; 4, -2, 1, -2; -2, 4, -2, 1; . Take antidiagonals of the matrices until all terms in each matrix are used. CROSSREFS Cf. A164092, A119633. Sequence in context: A274369 A055091 A014678 * A016533 A122915 A327193 Adjacent sequences:  A164513 A164514 A164515 * A164517 A164518 A164519 KEYWORD tabl,sign AUTHOR Gary W. Adamson, Aug 14 2009 STATUS approved

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Last modified January 20 14:24 EST 2020. Contains 331094 sequences. (Running on oeis4.)