A259911 Triangular array; row k shows the discriminant of the field of the number having purely periodic continued fraction with period (j,k+1-j), for j=1..k.

5, 12, 12, 21, 8, 21, 8, 60, 60, 8, 5, 24, 13, 24, 5, 60, 140, 12, 12, 140, 60, 77, 12, 285, 5, 285, 12, 77, 24, 28, 44, 120, 120, 44, 28, 24, 13, 5, 21, 168, 29, 168, 21, 5, 13, 140, 44, 168, 56, 1020, 1020, 56, 168, 44, 140, 165, 120, 93, 8, 1365, 40, 1365, 8, 93, 120, 165

Offset: 1

Author

Clark Kimberling, Jul 20 2015

Keywords

• nonn
• tabl
• easy

Examples

			First eight rows:
  5
  12    12
  21     8    21
   8    60    60     8
   5    24    13    24     5
  60   140    12    12   140    60
  77    12   285     5   285    12    77
  24    28    44   120   120    44    28    24
The number whose continued fraction is periodic with period (1,1) is the golden ratio, (1+sqrt(5))/2, so that the number in row 1 is 5.
As a square array A(n,k) read by antidiagonals, where A(n,k) corresponds to the continued fraction with pure period (n,k):
   5,  12,  21,   8,    5,   60,   77,  24, ...
  12,   8,  60,  24,  140,   12,   28,   5, ...
  21,  60,  13,  12,  285,   44,   21, 168, ...
   8,  24,  12,   5,  120,  168,   56,   8, ...
   5, 140, 285, 120,   29, 1020, 1365, 440, ...
  60,  12,  44, 168, 1020,   40, 1932, 156, ...
  77,  28,  21,  56, 1365, 1932,   53, 840, ...
  24,   5, 168,   8,  440,  156,  840,  17, ...
  ...
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A259912 (main diagonal of square array), A259913 (first column).

Programs

Mathematica

v = Table[FromContinuedFraction[{j, {k + 1 - j, j}}], {k, 1, 20}, {j, 1, k}];
TableForm[NumberFieldDiscriminant[v]]