David still has one more lesson to do. He wants to talk about Sequence Numbers that produce Fibonacci or Fibonacci like sequences. But he does not have that ready yet, so let me entertain you with some examples of his work in this area.
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Finding the Sequence Number for the
Fibonacci Sequence by
working backwards.
First go to the OEIS website, go to the
page for sequence A000045 (www.oeis.org/A00045 ), and copy all
of the terms of the Fibonacci Sequence that are 6 digits or less.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,
89, 144, 233, 377, 610, 987,
1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040
Then add zeros as needed so that each
term has 6 digits,
000000, 000001, 000001, 000002, 000003,
000005, 000008,
000013, 000021, 000034, 000055, 000089, 000144, 000233, 000377, 000610, 000987, 000597, 002584, 004181, 006765, 010946, 017711, 028657, 046368, 075025, 121393, 196418, 317811, 514229, 832040
Next take out the commas and spaces,
put a “0.” in front, and
take the inverse (raise it to the -1 power).
0.000000000001000001000002000003000005000008000 01300002100003400005500008900014400023300037700 06100009870015970025840041810067650109460177110 28657046368075025121393196418317811514229832040^-1
Take this over to the Wolfram Alpha
website to do our calculation (www.wolframalpha.com ). Just put the expression above into the blank box at the top of the page and hit “enter”.
The result is:
9.999989999990000000000000000000000000000000000 000000000... × 10^11
Which means that our sequence number should be 999998999999. This is a sequence number that
produces the Fibonacci Sequence in six digit strings. |
So now let's check it to see if it works. In the example below he made a change to the sequence number so that it would show terms using 12 digit strings instead of 6 digit strings.
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The Fibonacci Numbers or Fibonacci
Sequence: a(0) = 0, a(1) = 1, and when n>1 then a(n) = a(n-1) + a(n-2).
The Sequence Number for the Fibonacci
Sequence is 999,999,999,998,999,999,999,999.
1/999999999998999999999999 =
0.
000000000000 000000000001 000000000001 000000000002 000000000003 000000000005 000000000008 000000000013 000000000021 000000000034 000000000055 000000000089 000000000144 000000000233 000000000377 000000000610 000000000987 000000001597 000000002584 000000004181 000000006765 000000010946 000000017711 000000028657 000000046368 000000075025 000000121393 000000196418 000000317811 000000514229 000000832040 000001346269 000002178309 000003524578 000005702887 000009227465 000014930352 000024157817 000039088169 000063245986 000102334155 000165580141 000267914296 000433494437 000701408733 001134903170 001836311903 002971215073 004807526976 007778742049 012586269025 020365011074 032951280099 053316291173 086267571272 139583862445 225851433717 365435296162 591286729879 …
Written in 12 digit strings.
Accurate to the 57st non-zero
term.
Compare OEIS sequence A000045.
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Phillip
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