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A125076 Triangle with trigonometric properties, 6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

A125076 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 2Pi/Q.

The triangle may be constructed by considering the rows of A152063 as upward sloping diagonals. [From 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 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.

CROSSREFS

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

A152063 [From Gary W. Adamson, Nov 26 2008]

Sequence in context: A124019 A202179 A173588 * A220562 A215564 A189449

Adjacent sequences:  A125073 A125074 A125075 * A125077 A125078 A125079

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Nov 18 2006

STATUS

approved

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Last modified November 27 13:22 EST 2014. Contains 250196 sequences.