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A180063
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Pascal-like triangle with trigonometric properties, row sums = powers of 4; generated from shifted columns of triangle A180062.
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2
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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
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OFFSET
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0,3
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COMMENTS
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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, ...).
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LINKS
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FORMULA
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Given triangle A180062, shift columns upward so that the new triangle A180063 has (n+1) terms per row.
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EXAMPLE
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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;
...
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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