Replies to post #196450 on The Golden Lounge

[Stock]

144 is a dead Italian number. I do know that.

Phi and the Fibonacci Series

http://goldennumber.net/classic/fibonser.htm

Leonardo Fibonacci discovered the series which converges on phi

In the 12th century, Leonardo Fibonacci discovered a simple
numerical series that is the foundation for an incredible
mathematical relationship behind phi.

Starting with 0 and 1, each new number in the series is simply
the sum of the two before it.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 . . .

The ratio of each successive pair of numbers in the series approximates phi (1.618. . .) , as 5 divided by 3 is 1.666..., and 8 divided by 5 is 1.60.

The table below shows how the ratios of the successive numbers in the Fibonacci series quickly converge on Phi or (|) . After the 40th number in the series, the ratio is accurate to 15 decimal places.

1.618033988749895 . . .

 
   SEQUENCE   Fib #  Ratio to Prev. #  Difference from Phi     
 -------------------------------------------------------------------------------- 
 1             1       
 2             1 1.000000000000000 +0.618033988749895      
 3             2 2.000000000000000 -0.381966011250105      
 4             3 1.500000000000000 +0.118033988749895      
 5             5 1.666666666666667 -0.048632677916772      
 6             8 1.600000000000000 +0.018033988749895      
 7            13 1.625000000000000 -0.006966011250105      
 8            21 1.615384615384615 +0.002649373365279      
 9            34 1.619047619047619 -0.001013630297724      
 10           55 1.617647058823529 +0.000386929926365      
 11           89 1.618181818181818 -0.000147829431923      
 12          144 1.617977528089888 +0.000056460660007      
 13          233 1.618055555555556 -0.000021566805661      
 14          377 1.618025751072961 +0.000008237676933      
 15          610 1.618037135278515 -0.000003146528620      
 16          987 1.618032786885246 +0.000001201864649      
 17        1,597 1.618034447821682 -0.000000459071787      
 18        2,584 1.618033813400125 +0.000000175349770      
 19        4,181 1.618034055727554 -0.000000066977659      
 20        6,765 1.618033963166707 +0.000000025583188      
 21       10,946 1.618033998521803 -0.000000009771909      
 22       17,711 1.618033985017358 +0.000000003732537      
 23       28,657 1.618033990175597 -0.000000001425702      
 24       46,368 1.618033988205325 +0.000000000544570      
 25       75,025 1.618033988957902 -0.000000000208007      
 26      121,393 1.618033988670443 +0.000000000079452      
 27      196,418 1.618033988780243 -0.000000000030348      
 28      317,811 1.618033988738303 +0.000000000011592      
 29      514,229 1.618033988754323 -0.000000000004428      
 30      832,040 1.618033988748204 +0.000000000001691      
 31    1,346,269 1.618033988750541 -0.000000000000646      
 32    2,178,309 1.618033988749648 +0.000000000000247      
 33    3,524,578 1.618033988749989 -0.000000000000094      
 34    5,702,887 1.618033988749859 +0.000000000000036      
 35    9,227,465 1.618033988749909 -0.000000000000014      
 36   14,930,352 1.618033988749890 +0.000000000000005      
 37   24,157,817 1.618033988749897 -0.000000000000002      
 38   39,088,169 1.618033988749894 +0.000000000000001      
 39   63,245,986 1.618033988749895 -0.000000000000000      
 40  102,334,155 1.618033988749895 +0.000000000000000