The Compact Version of the Fibonacci Sequence

8/08/2015

8,082,015 is an 808,203-gonal and 2,694,006-gonal number.

The Compact Version of the Fibonacci Sequence:

You may remember that the Fibonacci starts with a zero, a one, and another one, and then each new number in the sequence is the sum of the previous two numbers:

1, 1, 2, 3, 5, 8, 13, 21, etc.

Instead of remember all of the terms (you can’t anyway, it is an infinite sequence), or having to calculate each term by hand, just remember the number 999,999,999,999,999,999,999,998,999, 999,999,999,999,999,999,999.

If you take this number, invert it (make it the denominator of a fraction with 1 as the numerator), and turn it into a decimal expansion, the decimal expansion will show the terms of the Fibonacci Sequence, written in 24 digit strings.

999,999,999,999,999,999,999,998,999,999,999,999,999,999,999,999 The inverse of this sequence numbers is: 1/999999999999999999999998999999999999999999999999 The digital expansion of the inverse is (please note that I have put blank spaces in to separate the terms and make it easier to read): 0. 000000000000000000000000  000000000000000000000001  000000000000000000000001  000000000000000000000002  000000000000000000000003  000000000000000000000005  000000000000000000000008  000000000000000000000013  000000000000000000000021  000000000000000000000034  000000000000000000000055  000000000000000000000089  000000000000000000000144  000000000000000000000233 000000000000000000000377  000000000000000000000610  000000000000000000000987  000000000000000000001597  000000000000000000002584  000000000000000000004181  000000000000000000006765  000000000000000000010946  000000000000000000017711  000000000000000000028657  000000000000000000046368  000000000000000000075025  000000000000000000121393  000000000000000000196418  000000000000000000317811  000000000000000000514229  000000000000000000832040  000000000000000001346269  000000000000000002178309  000000000000000003524578  000000000000000005702887  000000000000000009227465  000000000000000014930352  000000000000000024157817  000000000000000039088169  000000000000000063245986  000000000000000102334155  000000000000000165580141  000000000000000267914296  000000000000000433494437  000000000000000701408733  000000000000001134903170  000000000000001836311903  000000000000002971215073  000000000000004807526976  000000000000007778742049  000000000000012586269025  000000000000020365011074  000000000000032951280099  000000000000053316291173  000000000000086267571272  000000000000139583862445  000000000000225851433717  000000000000365435296162  000000000000591286729879  000000000000956722026041  000000000001548008755920  000000000002504730781961  000000000004052739537881  000000000006557470319842  000000000010610209857723  000000000017167680177565  000000000027777890035288  000000000044945570212853  000000000072723460248141  000000000117669030460994  000000000190392490709135  000000000308061521170129  000000000498454011879264  000000000806515533049393  000000001304969544928657  000000002111485077978050  000000003416454622906707  000000005527939700884757  000000008944394323791464  000000014472334024676221  000000023416728348467685  000000037889062373143906  000000061305790721611591  000000099194853094755497  000000160500643816367088  000000259695496911122585  000000420196140727489673  000000679891637638612258  000001100087778366101931  000001779979416004714189  000002880067194370816120  000004660046610375530309  000007540113804746346429  000012200160415121876738  000019740274219868223167  000031940434634990099905  000051680708854858323072  000083621143489848422977  000135301852344706746049  000218922995834555169026  000354224848179261915075  000573147844013817084101  000927372692193078999176  001500520536206896083277  002427893228399975082453  003928413764606871165730  006356306993006846248183  010284720757613717413913  016641027750620563662096  026925748508234281076009  043566776258854844738105  070492524767089125814114  114059301025943970552219  184551825793033096366333 298611126818977066918552  483162952612010163284885  …

It shows the first 115 non-zero terms of the Fibonacci Sequence.  If you want to check my math please see: http://oeis.org/A000045/b000045.txt .

David