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A125076
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Triangle with trigonometric properties, row sums = powers of 3.
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6
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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; internal format)
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OFFSET
| 1,3
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COMMENTS
| 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]
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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.
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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.
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CROSSREFS
| Cf. A125077, A125078.
A152063 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 26 2008]
Sequence in context: A124019 A202179 A173588 * A189449 A109533 A062705
Adjacent sequences: A125073 A125074 A125075 * A125077 A125078 A125079
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KEYWORD
| nonn,tabl
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 18 2006
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