GENERALIZED BINARY POINT SEQUENCES

We have already highlighted the remarkable point Fibonacci sequence that totals unity.

Thus once again

1 = .01 + .001 + .0010 + .00011 + .000101 + .0001000 + .00001101 + .000010101 + .0000100010 + …

Once again this is based on the format for the reciprocal of such sequences 100 - 10 - 1 = 1 (in binary terms).

This can be extended for the three-term, four-term and ultimately n-term cases.

Thus for the three term case, the sum of terms of the point sequence = the reciprocal of

100 - 10 - 1 - .1

Thus in binary the sum of terms = the reciprocal of .1 = 10.

(Incidentally in the 1-term case the formula for the sum of the point series = reciprocal of 100 - 10 ( = 10) which is .1

The relevant point series = .01 + .001 + .0001 + .00001 + .000001 + …

= .1)

The 3-term series is 1, 1, 2, 4, 7, 13, 24, 44, …

In binary 1 = 1, 2, = 10, 4 = 100, 7 = 111, 13 = 1101, 24 = 11000, 44 = 101100

Therefore 10 = .01 + .001 + .0010 + .00100 + .000111 + .0001101 + .00011000 + .000101100 + …

In the four-term case the sum of terms of the point sequence = the reciprocal of
100 - 10 - 1 - .1 - .01 = .01

Thus in binary the sum of terms = the reciprocal of .01 = 100

The four-term series is 1, 1, 2, 4, 8, 15, 29, 56,

In binary 1 = 1, 2 = 10, 4 = 100, 8 = 1000, 15 = 1111, 29 = 11101, 56 = 111000

Therefore

100 = .01 + .001 + .0010 + .00100 + .001000 + .0001111 + .00011101 + .000111000 +...

In like manner sum of the binary point series of the five-term case = 1000, the six-term = 10000,

The seven-term = 100000 etc

Thus the sum of the n-term case = 1 followed by 0 (repeated n - 2 times).

Once again the nature of these results in binary highlights the fact that the two-term case on which the Fibonacci sequence is based is simply a special case of a more generalized pattern of n-term behavior.

We can also generate results for the sums of binary point series where the sign of terms continually alternates.

Thus in the 3-term case the sum of the relevant point series = reciprocal of 100 + 10 - 1

= 101 (5 in denary terms)

Thus 1/101 = .01 - .001 + .0010 - .00100 + .000111 - .0001101 + .00011000 - .000101100 + …

In the 4-term case the sum of the relevant point series = reciprocal of 100 + 10 - 1 + .1

101.1 (5.5 in denary terms).

Thus 1/101.1 = .01 - .001 + .0010 - .00100 + .001000 - .0001111 + .00011101 - .000111000

The 5-term case = reciprocal of 101.09, six-term case 101.091, 7-term = 101.0909 etc.