

A125078


Fifth in an infinite set of generalized Pascal's triangles, with trigonometric properties.


3



1, 1, 4, 1, 5, 19, 1, 9, 24, 91, 1, 10, 63, 115, 436, 1, 14, 73, 397, 551, 2089, 1, 15, 132, 470, 2358, 2640, 10009, 1, 19, 147, 1043, 2828, 13482, 12649, 47956, 1, 20, 226, 1190, 7441, 16310, 75061, 60605, 229771
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OFFSET

1,3


COMMENTS

The triangle is the fifth in an infinite set of generalized Pascal's triangles constrained by two properties: row sums = powers of N and upward sloping diagonals solve for N + 2*Cos 2Pi/Q. Row sums are powers of 5. Right border (1, 4, 19, 91, 436...) = A004253. Next to right border (1, 5, 24, 115...) = A004254.


LINKS

Table of n, a(n) for n=1..45.


FORMULA

Upward sloping diagonals are derived from interleaved characteristic polynomials of two types of matrices, relating to odd and even polygons. Matrices with an eigenvalue 5 + 2*Cos 2Pi/Q, Q is odd, are of the form: all 1's in the super and subdiagonals and 4,5,5,5... in the main diagonal. Matrices (Q is even) are of the form: all 1's in the super and subdiagonals and 5,5,5... in the main diagonal.


EXAMPLE

First few rows of the triangle are:
1;
1, 4;
1, 5, 19;
1, 9, 24, 91;
1, 10, 63, 115, 436;
1, 14, 73, 397, 551, 2089;
1, 15, 132, 470, 2358, 2640, 10009;
...
The upward sloping diagonal (1, 14, 63, 91) is derived from the characteristic polynomial x^3  14x^2 + 63x  91 and relates to the Heptagon (Q=7) since a root = 6.24697960...= 5 + 2*Cos 2Pi/7. The corresponding matrix is [4, 1, 0; 1, 5, 1; 0, 1, 5]. The next upward sloping diagonal (1, 15, 73, 115) relates to the Octagon (Q=8) since a root = 6.41421356... = 5 + 2*Cos 2Pi/8. The corresponding matrix is [5, 1, 0; 1, 5, 1; 0, 1, 5].


CROSSREFS

Cf. A125076, A125078, A004253, A004254.
Sequence in context: A193955 A130746 A342925 * A087841 A194576 A229896
Adjacent sequences: A125075 A125076 A125077 * A125079 A125080 A125081


KEYWORD

nonn,tabl


AUTHOR

Gary W. Adamson, Nov 18 2006


STATUS

approved



