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

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