Examples of How to Find Sequence Numbers

9/22/2015

Examples of How to Find Sequence Numbers

Finding the Sequence Number for the Fibonacci Sequence (OEIS A000045) 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 (the first 30 terms). 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). 0.0000000000010000010000020000030000050000080000130 000210000340000550000890001440002330003770006100009 870015970025840041810067650109460177110286570463680 75025121393196418317811514229832040^-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: 9.9999899999900000000000000000000000000000000000000 00000... × 10^11 So our sequence number is 999998999999 (I hope)  Let’ check it  out at the Wolfram alpha website.

999,998,999,999 1/999998999999 = 0. 000000  000001  000001  000002  000003  000005  000008  000013  000021  000034  000055  000089  000144  000233  000377  000610  000987  001597  002584  004181  006765  010946  017711  028657  046368  075025  121393  196418  317811  514229  … The terms are written in 6 digit strings. The terms are accurate up to the 29th non-zero term.  The 30th term has an error because the 31st term is seven digits long and overlaps the 30th term, changing the 30th term to 832041 – which as I said, is not correct. This is a sequence number that produces the Fibonacci Sequence in six digit strings as predicted.  If you would like to see it produce more terms try 999,999,999,998,999,999,999,999 – it will produce terms that are 12 digits long.

Finding the Sequence Number for the Pentanacci Numbers (OEIS A001591) by working backwards. Copy the terms of the Pentanacci Sequence up through all of the six digit terms. 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 Add zeros to make every term six digits, take out the commas and spaces, add a “0.” on the front, and take the inverse (tac a “^-1” on the back end). 0.0000000000000000000000000000010000010000020000040 000080000160000310000610001200002360004640009120017 930035250069300136240267840526561035192035134000967 86568^-1 Do the calculation. 9.9999899999899999899999899999900000000000000000000 00000... × 10^29 Which equals: 999998999998999998999998999999.  This sequence number produces the Pentanacci Sequence written in six digit strings.  Other sequence numbers related to this one will produce longer string and more terms if needed or desired.  999999999998999999999998999999999998999999999998999 999999999 will produce terms in 12 digit terms, about twice as many terms as the previous number.

Fibonacci Bisection, (OEIS A001906) Every Other Fibonacci Numbers (the even numbered terms) First 16 terms are: 0, 1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765, 17711, 46368, 121393, 317811, 832040 Turn each term into a six digit string, take out the commas and the spaces, put a “0.” in the front and a “^-1” in the back.  This will give you: 0.000000000001000003000008000021000055000144000 37700098700258400676501771104636812139331781183 2040^-1 Plug the above number into Wolfram Alpha and you will get the following result: 9.999970000010000000000000000000000000000000000 000000000... × 10^11 It looks like the sequence number will be 999,997,000,001. So let’s check it out – see if it works. 1/999997000001 = 0. 000000  000001  000003  000008  000021  000055  000144  000377  000987  002584  006765  017711  046368  121393  317811  … Terms are written in six digit strings. Terms are accurate up to the 14th non-zero term. Additional terms can be displayed by showing 12 digit terms using the sequence number: 999,999,999,997,000,000,000,001.

The Other Every Other Fibonacci Numbers, A001519 OEIS describes this sequence as a bisection of the Fibonacci Sequence, listing the odd terms – but they include both 1s in this sequence and one of these must be an even numbered term.  I have not yet been able to find a Sequence Number that works for this sequence.  I suspect that there is no Sequence number that works for this sequence, but I am not closing the door on this one.  I really would like to find it if one exists.

This method of finding sequence numbers can be used on other number sequences also.  However, not all sequences have a sequence number that will produce them.  Very few do.  But you won’t know unless you try.

In the past though, I have found that I can often find sequence numbers for counting sequences, multiplying sequences, power sequence and Fibonacci like sequences (along with a few miscellaneous sequences).

I am working on a lesson to explain how the Fibonacci like sequences work.  I was not satisfied with my first draft.  I am working on a re-write that will break it up into two lessons – and have a surprise ending.

David