A125076 Triangle with trigonometric properties,

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, 1, 11, 51, 183, 300, 654, 377, 610, 1, 12, 74, 234, 717, 954, 1985, 987, 1597, 1, 14, 86, 394, 951, 2622

Offset

1,3

Comments

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).

The triangle may be constructed by considering the rows of A152063 as upward sloping diagonals. - Gary W. Adamson, Nov 26 2008

Links

Table of n, a(n) for n=1..51.

Formula

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.

Example

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(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).

Crossrefs

Cf. A125077, A125078, A000244 (row sums).

Cf. A152063. - Gary W. Adamson, Nov 26 2008

Sequence in context: A297749 A173588 A286942 * A220562 A215564 A189449

Adjacent sequences: A125073 A125074 A125075 * A125077 A125078 A125079

Keyword

nonn,tabl

Author

Gary W. Adamson, Nov 18 2006

Status

approved