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%I #8 May 25 2019 22:15:10
%S 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,
%T 859,780,2131,1,15,93,499,1091,3821,2911,7953,1,16,140,592,2774,4912,
%U 16556,10864,29681,1,19,156,1044,3366,14418,21468,70356,40545,110771
%N Pascal-like triangle with trigonometric properties, row sums = powers of 4; generated from shifted columns of triangle A180062.
%C Row sums = powers of 4, A000302: (1, 4, 16, 64, ...).
%C Rightmost terms of each row = A001835: (1, 3, 11, 41, 153, 571, ...).
%C 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,...
%C 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,...).
%C 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.
%C 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, ...).
%C 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, ...).
%F Given triangle A180062, shift columns upward so that the new triangle A180063 has (n+1) terms per row.
%e First few rows of the triangle:
%e 1;
%e 1, 3;
%e 1, 4, 11;
%e 1, 7, 15, 41;
%e 1, 8, 38, 56, 153;
%e 1, 11, 46, 186, 209, 571;
%e 1, 12, 81, 232, 859, 780, 2131;
%e 1, 15, 93, 499, 1091, 3821, 2911, 7953;
%e 1, 16, 140, 592, 2774, 4912, 16556, 10864, 29681;
%e 1, 19, 156, 1044, 3366, 14418, 21468, 70356, 40545, 110771;
%e ...
%Y Cf. A007318, A180062, A003835, A000302.
%K nonn,tabl
%O 0,3
%A _Gary W. Adamson_, Aug 08 2010
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