A125078 -id:A125078 - cp's OEIS frontend

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

Gary W. Adamson, Nov 18 2006

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

			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).

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

Cf. A152063. - Gary W. Adamson, 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.

A125077 #4 in an infinite set of generalized Pascal's triangles with trigonometric properties.

Original entry on oeis.org

1, 1, 3, 1, 4, 11, 1, 7, 15, 41, 1, 8, 38, 56, 153, 1, 11, 46, 186, 209, 571, 1, 12, 81, 232, 859, 780, 2131, 1, 15, 93, 499, 1091, 3821, 7953, 1, 16, 140, 592, 2774, 4912, 16556, 10864, 29681, 1, 19, 156, 1044, 3366

Offset: 1

Gary W. Adamson, Nov 18 2006

Row sums are powers of 4. The triangle is #4 in an infinite of generalized Pascal's triangles constrained by two rules: row sums are powers of N and upward sloping diagonals (as coefficients to polynomials with alternating signs) have roots N + 2*cos(2*Pi/Q).

Right border, A001835, and next to right border, A001353 = bisections of denominator of continued fraction [1, 2, 1, 2, 1, 2, 1, 2]; i.e., bisection of A002530. - Gary W. Adamson, Jun 21 2009

			First few rows of the triangle are:
  1;
  1,  3;
  1,  4, 11;
  1,  7, 15,  41;
  1,  8, 38,  56, 153;
  1, 11, 46, 186, 209, 571;
  1, 12, 81, 232, 859, 780, 2131;
  ...
The upward-sloping diagonal (1, 11, 38, 41) relates to the heptagon and in the form x^3 - 11x^2 + 38x - 41 has a root 5.24697960... = 4 + 2*cos(2*Pi/7). The corresponding matrix is [3, 1, 0; 1, 4, 1; 0, 1, 4]. The next upward-sloping diagonal relates to the octagon, with a characteristic polynomial x^3 - 12x^2 + 46x - 56 and a root 5.414213562... = 4 + 2*cos(2*Pi/8). The corresponding matrix is [4, 1, 0; 1, 4, 1; 0, 1, 4].

Cf. A125076, A125078.

Cf. A001835, A001353. - Gary W. Adamson, Jun 21 2009

Upward-sloping diagonals of the triangle are derived from (alternating) characteristic polynomials of two types of matrices: those of the form: (all 1's in the super and subdiagonals and 3,4,4,4,... in the main diagonal) and (all 1's in the super and subdiagonals and 4,4,4,... in the main diagonal.

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A125076 Triangle with trigonometric properties.

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