N-TERM GENERALIZATION OF FIBONACCI SEQUENCE
 
 
 
The Fibonacci sequence is based on continuously adding the last two terms to generate the next terms (starting with 0 and 1).

However the very notion of the Fibonacci can be generalized by continuously adding the last 3, 4, 5 …..n terms to generate important new sequences with fascinating properties..

Thus (strictly starting with 0, 0, 1) when we continuously add the last three terms to generate the next term, the following sequence emerges

0,  0,  1,  1,  2,  4,  7,  13,  24,  44,  81,  149,  274,  504,  927,  1705,  3136,  5768, 10609,  19513,  35890,  66012,  121415,  223317,  410744,  755476,  1389537, 2555757,  4700770,  8646064,…

The ratio here is 1.839286755…

This is the positive real valued solution to the equation

x3 - x2 - x - 1 = 0; here 1/x = x2 - x - 1.

Phi (1.6180339887…) which is the corresponding ratio in the two-term case is the solution to positive real solution to the equation

x2 - x - 1 = 0; here 1/x = x - 1.

This result can be perfectly generalized. Thus when the last n terms of the sequence are continuously added to generate the next term, the resulting ratio that emerges from dividing the last by the second last term in the sequence approximates to the positive real valued solution of the polynomial equation

xn - xn - 1 - xn - 2 ,…...., - x2 - x - 1 = 0. Here 1/x = xn - 1 - xn - 2 ,……, - x2 - x - 1.

So the Fibonacci sequence therefore represents a special case of a more generalized numerical pattern where n = 2. The ratio that results (phi) is therefore also a special case of a more generalized set of ratios (where n = 2).

Therefore I propose that we generalize this set of ratios labeling each one phi with a corresponding subscript (representing the value of n).

  • Thus phi1 is the ratio that results that starts from 1 combining 1 term at a time.
    • It therefore results in the series 1, 1, 1, 1, 1, 1, …….

    Therefore phi1 = 1 which is the (positive) real solution to the equation x - 1 = 0; here 1/x = 1.
     
     

  • Phi2 (which is popularly known as just phi) is the ratio that results from – starting with 0, 1 - combining (the last) two terms to generate the next term.


    • It results in the well known Fibonacci sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ….

    The ratio phi2 (i.e. phi) is 1.6180339887…

    Again this is the positive real valued solution to the equation x2 - x - 1 = 0; here 1/x = x - 1.
     
     

  • Phi3, as we have seen is the ratio that results from - starting with 0, 0, 1 – then
    • continually combining (the last) three terms to generate the next term.

    It results in the series 0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274,….

    The ratio phi3 is 1.839286755… and is the positive real valued solution to the equation x3 - x2 - x - 1 = 0; here 1/x = x2 - x - 1.
     
     

  • Phi4, is the ratio that results from - starting with 0, 0, 0, 1 – then
    • continually combining (the last) four terms to generate the next term.

    It results in the series

    0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536 10671, 20569, 39648, 76424, 147312, 283953, 547337,…

    The ratio is 1.9275619…and is the positive real-valued solution to the equation

    x4 - x3 – x2 - x – 1 = 0; here 1/x = x3 – x2 - x – 1.
     
     

  • Phi5, is the ratio that results from - starting with 0, 0, 0, 0, 1 – then
    • continually combining (the last) five terms to generate the next term.

    It results in the series

    0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, 26784, 52656, 103519, 203513, 400096, 786568, 1546352,…

    The ratio is 1.9659482… and is the positive real-valued solution to the equation

    x5 - x4 - x3 – x2 - x – 1 = 0; here 1/x = x4 - x3 – x2 - x – 1.
     
     

  • Phi6, is the ratio that results from - starting with 0, 0, 0, 0, 0, 1 – then
    • continually combining (the last) six terms to generate the next term.

    It results in the series 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 482, 966, 1916, 3800, 7537, 14949, 29650, 58818, 116670, 231424, 459048, 910559, 1806169,

    The ratio is 1.98358… and is the positive real-valued solution to

    x6 - x5 - x4 - x3 – x2 - x – 1 = 0; here 1/x = x5 -x4 - x3 – x2 - x – 1.
     
     

  • So generally phin is the ratio that results from – starting with 0 (repeated n – 1 times), 1 – then continually combining the last n terms to generate the next term.
  • Briefly, the 7-term case results in the series
    • 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808,..

    Phi7 is 1.9919…

    The 8-term case results in the series

    0, 0, 0, 0, 0, 0, 0, 1 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080, 16128,..

    Phi8 is 1.99604 (approx)
     
     

    The 9-term case results in the series

    0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272,…

    Phi9 is 1.998 (approx)
     
     

    The 10-term case results in the series

    0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8. 16, 32, 64, 128, 256 512, 1023, 2045, 4088, 8172, 16336, 32656, 65280,..

    Phi10 is 1.99902 (approx)
     
     

    Finally to illustrate the 12-term case results in the series

    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8. 16, 32, 64, 128, 256 512, 1024, 2048, 4095,

    8189, 16376, 32748, 65488, 130960, 261888,…

    Phi12 is 1.999756 (approx)
     
     

    Thus we can generate an infinite series of ratios for phin ranging from 1 to 2 as the number of (final) terms combined in sequence ranges from n = 1 to n = ¥ .

    Once again the Fibonacci Sequence with its resulting approximating ratio for phi represents a special case where n = 2.
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     

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