PELL AND HALF COMPANION PELL SEQUENCES
Pell and half companion Pell numbers can be generated a using a recursion method. Starting with 1/1 and continuing forward by plugging each successive term back into the fold, adding denominator to numerator to generate the next denominator, then adding the two denominators together to generate the next numerator.
TRIANGULAR SQUARE AND TRIANGULAR PRONIC SEQUENCES
The intersection of triangular and square numbers can be expressed as a ratio between the mth triangular and nth square ordinal numbers, this is true for the intersection of triangular and pronic numbers as well. This proportion can be generated a using a recursion method, starting with a seed and plugging each successive term back into the fold.
INTERLOCKING SEQUENCES
Composite Pell and Half-companion Pell sequences alternate between ratios that overestimate and underestimate the square root of 2. The triangular square progression interlocks additively with the series of ratios that underestimate the square root of 2. The triangular pronic progression interlocks additively with the series of ratios that overestimates.
INTERSECTION
Above are the first 696 triangular numbers inscribed within 492 concentric even-sided polygons. Triangular numbers intersect pronic and square numbers along the x-axis. The subsequent coordinates form a series of line segments whose length from the center is a successive Pell number.