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 A180063 Pascal-like triangle with trigonometric properties, row sums = powers of 4; generated from shifted columns of triangle A180062. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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 3x3 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,...). LINKS FORMULA Given triangle A180062, shift columns upward so that the new triangle A180063 has (n+1) terms per row. EXAMPLE 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 Sequence in context: A137405 A121922 A054631 * A125077 A065253 A010756 Adjacent sequences:  A180060 A180061 A180062 * A180064 A180065 A180066 KEYWORD nonn,tabf AUTHOR Gary W. Adamson, Aug 08 2010 STATUS approved

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