%I A125077
%S A125077 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 A125077 859,780,2131,1,15,93,499,1091,3821,7953,1,16,140,592,2774,4912,16556,
%U A125077 10864,29681,1,19,156,1044,3366
%N A125077 #4 in an infinite set of generalized Pascal's triangles with trigonometric 
               properties.
%C A125077 Row sums are powers of 4. The triangle is #4 in an infinite of generalized 
               Pascal's triangles constrained by two rules: row sums are powers 
               of N and upward sloping diagonals (as coefficients to polynomials 
               with alternating signs) have roots N + 2*Cos 2Pi/Q.
%C A125077 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 21 2009: 
               (Start)
%C A125077 Right border, A001835: (1, 3, 11, 41, 153,...); and next to right border,
%C A125077 A001353: (1, 4, 15, 56, 209, 780,...) = bisections of denominator of
%C A125077 continued fraction [1, 2, 1, 2, 1, 2, 1, 2]; i.e. bisection of
%C A125077 [1, 3, 4, 11, 15, 41, 56,...]. (End)
%F A125077 Upward sloping diagonals of the triangle are derived from (alternating) 
               characteristic polynomials of two types of matrices: those of the 
               form: (all 1's in the super and subdiagonals and 3,4,4,4... in the 
               main diagonal) and (all 1's in the super and subdiagonals and 4,4,
               4... in the main diagonal.
%e A125077 First few rows of the triangle are:
%e A125077 1;
%e A125077 1, 3;
%e A125077 1, 4, 11;
%e A125077 1, 7, 15, 41;
%e A125077 1, 8, 38, 56, 153;
%e A125077 1, 11, 46, 186, 209, 571;
%e A125077 1, 12, 81, 232, 859, 780, 2131;
%e A125077 ...
%e A125077 The upward sloping diagonal (1, 11, 38, 41) relates to the Heptagon and 
               in the form x^3 - 11x^2 + 38x - 41 has a root 5.24697960...= 4 + 
               2*Cos 2Pi/7. The corresponding matrix is [3, 1, 0; 1, 4, 1; 0, 1, 
               4]. The next upward sloping diagonal relates to the Octagon, with 
               a characteristic polynomial x^3 - 12x^2 + 46x - 56 and a root 5.414213562... 
               = 4 + 2*Cos 2Pi/8. The corresponding matrix is [4, 1, 0; 1, 4, 1; 
               0, 1, 4].
%Y A125077 Cf. A125076, A125078.
%Y A125077 A001835, A001353 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 21 
               2009]
%Y A125077 Sequence in context: A137405 A121922 A054631 this_sequence A065253 A010756 
               A153278
%Y A125077 Adjacent sequences: A125074 A125075 A125076 this_sequence A125078 A125079 
               A125080
%K A125077 nonn,tabl
%O A125077 1,3
%A A125077 Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 18 2006