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%I #16 Jun 05 2021 08:41:02
%S 1,1,2,1,3,5,1,5,8,13,1,6,19,21,34,1,8,25,65,55,89,1,9,42,90,210,144,
%T 233,1,11,51,183,300,654,377,610,1,12,74,234,717,954,1985,987,1597,1,
%U 14,86,394,951,2622
%N Triangle with trigonometric properties,
%C This triangle is #3 in an infinite set, where Pascal's triangle = #2. Generally, the infinite set is constrained by two properties: For triangle N, row sums are powers of N and upward sloping diagonals have roots equal to N + 2*cos(2*Pi/Q).
%C The triangle may be constructed by considering the rows of A152063 as upward sloping diagonals. - _Gary W. Adamson_, Nov 26 2008
%F Upward sloping diagonals are alternating (unsigned) characteristic polynomial coefficients of two forms of matrices: all 1's in the super and subdiagonals and (2,3,3,3,...) in the main diagonal and the other form all 1's in the super and subdiagonals and (3,3,3,...) in the main diagonal.
%e First few rows of the triangle are:
%e 1;
%e 1, 2;
%e 1, 3, 5;
%e 1, 5, 8, 13;
%e 1, 6, 19, 21, 34;
%e 1, 8, 25, 65, 55, 89;
%e 1, 9, 42, 90, 210, 144, 233;
%e ...
%e For example, the upward-sloping diagonal (1, 8, 19, 13) is derived from x^3 - 8x^2 + 19x - 13, characteristic polynomial of the 3 X 3 matrix [2, 1, 0; 1, 3, 1;, 0, 1, 3], having an eigenvalue of 3 + 2*cos(2*Pi/7). The next upward-sloping diagonal is (1, 9, 25, 21), derived from the characteristic polynomial x^3 - 9x^2 + 25x - 21 and the matrix [3, 1, 0; 1, 3, 1; 0, 1, 3]. An eigenvalue of this matrix and a root of the corresponding characteristic polynomial is 4.414213562... = 3 + 2*cos(2*Pi/8).
%Y Cf. A125077, A125078, A000244 (row sums).
%Y Cf. A152063. - _Gary W. Adamson_, Nov 26 2008
%K nonn,tabl
%O 1,3
%A _Gary W. Adamson_, Nov 18 2006
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