cp's OEIS Frontend

A125076 Triangle with trigonometric properties.

%I A125076 #18 Jun 01 2025 16:59:04
%S A125076 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 A125076 233,1,11,51,183,300,654,377,610,1,12,74,234,717,954,1985,987,1597,1,
%U A125076 14,86,394,951,2622
%N A125076 Triangle with trigonometric properties.
%C A125076 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 A125076 The triangle may be constructed by considering the rows of A152063 as upward sloping diagonals. - _Gary W. Adamson_, Nov 26 2008
%F A125076 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 A125076 First few rows of the triangle are:
%e A125076   1;
%e A125076   1, 2;
%e A125076   1, 3,  5;
%e A125076   1, 5,  8, 13;
%e A125076   1, 6, 19, 21,  34;
%e A125076   1, 8, 25, 65,  55,  89;
%e A125076   1, 9, 42, 90, 210, 144, 233;
%e A125076   ...
%e A125076 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 A125076 Cf. A125077, A125078, A000244 (row sums).
%Y A125076 Cf. A152063. - _Gary W. Adamson_, Nov 26 2008
%K A125076 nonn,tabl
%O A125076 1,3
%A A125076 _Gary W. Adamson_, Nov 18 2006