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%I #8 Feb 23 2013 12:42:24
%S 1,1,4,1,5,19,1,9,24,91,1,10,63,115,436,1,14,73,397,551,2089,1,15,132,
%T 470,2358,2640,10009,1,19,147,1043,2828,13482,12649,47956,1,20,226,
%U 1190,7441,16310,75061,60605,229771
%N Fifth in an infinite set of generalized Pascal's triangles, with trigonometric properties.
%C The triangle is the fifth in an infinite set of generalized Pascal's triangles constrained by two properties: row sums = powers of N and upward sloping diagonals solve for N + 2*Cos 2Pi/Q. Row sums are powers of 5. Right border (1, 4, 19, 91, 436...) = A004253. Next to right border (1, 5, 24, 115...) = A004254.
%F Upward sloping diagonals are derived from interleaved characteristic polynomials of two types of matrices, relating to odd and even polygons. Matrices with an eigenvalue 5 + 2*Cos 2Pi/Q, Q is odd, are of the form: all 1's in the super and subdiagonals and 4,5,5,5... in the main diagonal. Matrices (Q is even) are of the form: all 1's in the super and subdiagonals and 5,5,5... in the main diagonal.
%e First few rows of the triangle are:
%e 1;
%e 1, 4;
%e 1, 5, 19;
%e 1, 9, 24, 91;
%e 1, 10, 63, 115, 436;
%e 1, 14, 73, 397, 551, 2089;
%e 1, 15, 132, 470, 2358, 2640, 10009;
%e ...
%e The upward sloping diagonal (1, 14, 63, 91) is derived from the characteristic polynomial x^3 - 14x^2 + 63x - 91 and relates to the Heptagon (Q=7) since a root = 6.24697960...= 5 + 2*Cos 2Pi/7. The corresponding matrix is [4, 1, 0; 1, 5, 1; 0, 1, 5]. The next upward sloping diagonal (1, 15, 73, 115) relates to the Octagon (Q=8) since a root = 6.41421356... = 5 + 2*Cos 2Pi/8. The corresponding matrix is [5, 1, 0; 1, 5, 1; 0, 1, 5].
%Y Cf. A125076, A125078, A004253, A004254.
%K nonn,tabl
%O 1,3
%A _Gary W. Adamson_, Nov 18 2006
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