1,3

Row sums are powers of 3. A125076 is #3 in an infinite set, where Pascal's triangle = #2. Generally, the infinite set is constrined by two properties: For triangle N, row sums are powers of N and upward sloping diagonals have roots equal to N + 2*Cos 2Pi/Q.

The triangle may be constructed by considering the rows of A152063 as upward sloping diagonals. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 26 2008]

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.

First few rows of the triangle are:

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, 233;

...

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 2Pi/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 2Pi/8.

Cf. A125077, A125078.

A152063 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 26 2008]

Sequence in context: A110197 A124819 A124019 this_sequence A109533 A062705 A059234

Adjacent sequences: A125073 A125074 A125075 this_sequence A125077 A125078 A125079

nonn,tabl

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