login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A125076 Triangle with trigonometric properties, 6

%I

%S 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,

%T 233,1,11,51,183,300,654,377,610,1,12,74,234,717,954,1985,987,1597,1,

%U 14,86,394,951,2622

%N Triangle with trigonometric properties,

%C 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.

%C The triangle may be constructed by considering the rows of A152063 as upward sloping diagonals. [From _Gary W. Adamson_, Nov 26 2008]

%F 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.

%e First few rows of the triangle are:

%e 1;

%e 1, 2;

%e 1, 3, 5;

%e 1, 5, 8, 13;

%e 1, 6, 19, 21, 34;

%e 1, 8, 25, 65, 55, 89;

%e 1, 9, 42, 90, 210, 144, 233;

%e ...

%e 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.

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

%Y A152063 [From _Gary W. Adamson_, Nov 26 2008]

%K nonn,tabl

%O 1,3

%A _Gary W. Adamson_, Nov 18 2006

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 7 13:12 EDT 2021. Contains 343650 sequences. (Running on oeis4.)