1,3

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.

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.

First few rows of the triangle are:

1;

1, 4;

1, 5, 19;

1, 9, 24, 91;

1, 10, 63, 115, 436;

1, 14, 73, 397, 551, 2089;

1, 15, 132, 470, 2358, 2640, 10009;

...

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].

Cf. A125076, A125078, A004253, A004254.

Adjacent sequences: A125075 A125076 A125077 this_sequence A125079 A125080 A125081

Sequence in context: A100279 A132379 A130746 this_sequence A087841 A127140 A010642

nonn,tabl

Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 18 2006