1,3

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.

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 21 2009: (Start)

Right border, A001835: (1, 3, 11, 41, 153,...); and next to right border,

A001353: (1, 4, 15, 56, 209, 780,...) = bisections of denominator of

continued fraction [1, 2, 1, 2, 1, 2, 1, 2]; i.e. bisection of

[1, 3, 4, 11, 15, 41, 56,...]. (End)

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.

First few rows of the triangle are:

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;

...

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].

Cf. A125076, A125078.

A001835, A001353 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 21 2009]

Sequence in context: A137405 A121922 A054631 this_sequence A065253 A010756 A153278

Adjacent sequences: A125074 A125075 A125076 this_sequence A125078 A125079 A125080

nonn,tabl

Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 18 2006