Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

In the quintuple series, 5, 25, 125, 625, kc. it will be found
that each term of the series exceeds the quadruple of the sum
of its parts, by unity.
In the septuple series, each term exceeds the sextuple of the
sum of its parts, by unity.
In the noncuple series, each term exceeds the octuple of the
sum of its parts, by unity. And thus in all series formed by
the multiplication of odd numbers, each term will exceed the
sum arising from the multiplication of its parts by the odd
number, by unity.
If to each term of the
series -.----.-.-.-.-..---.--.- 12 4 8 16 32 64 128 256
6 is added, viz. -..--..-.... 6 6 6 6 6 6 6 6 6
the sums will be .-..--...- 7 8 10 14 22 38 70 134 262
And if to the series ..-. 1 3 9 27 81 243 729 2187 6561
be added -..-....--.......------- 3 6 6 6 6 6 6 6 6
the sums will be .---.--.-.-. 4 9 15 33 87 249 735 2193 6567, kc.
In which it is remarkable, that the aggregates of the parts of
the five sums 8, 10, 14, 22, 38, are 7, 8, 10, 14, 32; and of the
parts of the five sums 9, 15, 33, 87, 249, are 4, 9, 15, 33, 87.
The aggregates also of the parts of the sums 134, 262, are 70,
134; but this will not be the case with the sums beyond 262,
if the terms of the duple series are continued, and 6 is added
to them. Nor will it be the case with the sums of the triple
series beyond 249.
CHAPTER
On the series of terms mjing from the multiplicution of evenly-even
numbers, by the sums produced by the addition of