 ## Modified Fibonacci Series A “modified” series of Fibonacci Numbers can be easily had by using starting numbers other than 0 and 1.  For example, we can write a whole series of modified Fibonacci series by using as the first numbers, 1 and another integer.  This is shown in Table 1.  In fact, we can also use non-integer numbers (as in the so-called “crossing sequence” in Golden Mean Mathematics, where we used 1 and Ö5).  In terms of the ratio of adjacent numbers always approaching the Golden Mean (F or f), everything seems to work!

Table 1

1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349...

1, 4, 5, 9, 14, 23, 37, 60, 97, 157, 254, 411, 665, 1076, 1741, 2817, 4558, 7375, 11,933...

1, 5, 6, 11, 17, 28, 45, 73, 118, 191, 309, 500, 809, 1309, 2118, 3427, 5545, 8972...

1, 6, 7, 13, 20, 33, 53, 86, 139, 225, 364, 589, 953, 1542, 2495, 4037, 6532, 10,569...

1, 7, 8, 15, 23, 38, 61, 99, 160, 259, 419, 678, 1097, 1775, 2872, 4647, 7519, 12,166...

A significant result is: Any series which uses the methods of adding the last two numbers to obtain the next in the series, will always have as a limit of the ratio of those last two numbers, F and f.  Obviously, if the numbers are squared, cubed and so forth -- i.e. any of the numbers taken to the nth power -- we will have as limits, Fn and fn.  More importantly, if there is a separation between the numbers (where the numbers between the ratio numbers total n), we obtain the equivalent limits of  Fn+1 and fn+1.  For example, dividing 2817 by 11,933 yields 0.2361, which in turn equals 0.618... to the 3rd (2+1) power -- the 2 representing the separation of the two numbers, 4558 and 7375.

No.

For in continuing the ialexandrian tradition of observing strange phenomena, we might note that the 12th number of each sequence (not counting the zero in the original Fibonacci Series and noted in bold in Table 1), i.e. 233, 322, 411, 500, 589, 678... all differ by 89 between adjacent sequences.  89 is, of course, the 10th number of the original Fibonacci series.  Meanwhile, the 17th numbers, 2584, 3571, 4558, 5545, 6532, 7519... all differ by 987, the 15th number of the original Fibonacci series.  Not surprisingly, the general rule is that the nth number of all of the different, modified Fibonacci series, all differ between the adjacent series by the (n-2)th number of the original Fibonacci series.

We can also match a Fibonacci series with its cumulative sequence.  Table 2 shows two examples: the traditional Fibonacci series and its cumulative totals on the second line, as well as the 1-3 sequences matched to its cumulative sequence.

Table 2

1, 2, 3,  5,   8, 13, 21, 34,   55,   89, 144, 233, 377,   610,   987, 1397, 2584, 4181...

1, 3, 6,11, 19, 32, 53, 87, 142, 231, 375, 608, 985, 1595, 2582, 3979, 6563, 10,744...

1, 3, 4,   7, 11, 18, 29,   47,  76, 123, 199, 322,   521,  843, 1364, 2207, 3571, 5778...

1, 4, 8, 15, 26, 44, 73, 120, 196, 319, 518, 840, 1361, 2204, 3568, 5775, 9346,15,124...

The cumulative sequences also approach F and f!  But you probably guessed that, right?

What you might not have guessed is the manner in which the nth number in the cumulative sequence is less than the n+2 number in the regular sequence by the second number in the regular sequence.  Really.

Returning to the “crossing” Fibonacci series defined in Golden Mean Mathematics by Fn + fn and Fn - fn, we note that the whole numbers of the opposing cross sequence of numbers are also exact square roots.  Comparing the two series...

5, 20, 45, 125, 320, 845, 2205, 5780, 15125, 39605, 103680, 271445, 710645, 1860500

9,16, 49, 121, 324, 841, 2209, 5776, 15129, 39601, 103684, 271441, 710,649, 1860496

We note that each term differs from the corresponding number in the other series by a factor of 4 (or 22), with the two series interweaving between themselves.  The two series can be said to be based on Ö5 = F + f and Ö9 = F - f + 2.  If we then compare any two adjacent numbers in the sequence, we see that the quotients of adjacent numbers approach f2 and F2, and the quotients of two non-adjacent numbers approach fn+1 and Fn+1 where n is the count of numbers in the series between the numbers being divided.  Amazing!

In summary, all of the Fibonacci numbers, taken in a host of different combinations and different sequences, all connect directly through the Golden Means, turning into and connecting with themselves like fractals in Chaos Theory.

Mathematics can truly be philosophical -- as in Sacred Geometry, Transcendental Numbers, and F-lo-sophia (to name just a few).

Or press on to Sacred Geometry, Music, Connective Physics, and Creating Reality.

But whatever you do, don’t stop now!

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