A180063 Pascal-like triangle with trigonometric properties, row sums = powers of 4; generated from shifted columns of triangle A180062.

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, 2911, 7953, 1, 16, 140, 592, 2774, 4912, 16556, 10864, 29681, 1, 19, 156, 1044, 3366, 14418, 21468, 70356, 40545, 110771

Offset: 0

Author

Gary W. Adamson, Aug 08 2010

Keywords

• nonn
• tabl

Comments

Row sums = powers of 4, A000302: (1, 4, 16, 64, ...).

Rightmost terms of each row = A001835: (1, 3, 11, 41, 153, 571, ...).

A180063 may be considered N=4 in an infinite set of Pascal-like triangles generated from variants of the Cartan matrix. Such triangles have trigonometric properties in charpolys being the upward sloping diagonals (cf. triangle A180062 = upward sloping diagonals of A180063); as well as row sums = powers of 2,3,4,...

Triangle A125076 = N=3, with row sums powers of 3; (if the original Pascal's triangle A007318 is considered N=2). To generate the infinite set of these Pascal-like triangles we use Cartan matrix variants with (1's in the super and subdiagonals) and (N-1),N,N,N,... as the main diagonal, alternating with (N,N,N,...).

For example, in the current N=4 triangle, row 7 of A180062 relates to the Heptagon and is generated from the 3 X 3 matrix [3,1,0; 1,4,1; 0,1,4], charpoly x^3 - 11x^2 + 38x - 41. Thus row 7 of triangle A180062 = (1, 11, 38, 41) = an upward sloping diagonal of triangle A180063.

The upward sloping diagonals of the infinite set of Pascal-like triangles = denominators in continued fraction convergents to [1,N,1,N,1,N,...] such that Pascal's triangle (N=2, A007318) has the Fibonacci terms generated from [1,1,1,...]. Similarly, for the case (N=3, triangle A125076), the upward sloping diagonals = row terms of triangle A152063 and are denominators in convergents to [1,2,1,2,1,2,...] = (1, 3, 4, 11, 15, ...).

Triangle A180063 is generated from upward sloping diagonals of triangle A180062, sums found as denominators in [1,3,1,3,1,3,...] = (1, 4, 5, 19, ...).

Examples

			First few rows of the triangle:
  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,  2911,  7953;
  1, 16, 140,  592, 2774,  4912, 16556, 10864, 29681;
  1, 19, 156, 1044, 3366, 14418, 21468, 70356, 40545, 110771;
  ...
		

Crossrefs

Cf. A007318, A180062, A003835, A000302.

Formula

Given triangle A180062, shift columns upward so that the new triangle A180063 has (n+1) terms per row.