Not Sequence Numbers – But Kind of Like Sequence Numbers
This does not fit the definition of a sequence number. However, it does have an interesting “sequence number-ish” result, and may help someone else find a sequence number for the Lucas Sequence.
The Lucas Sequence
The Lucas Sequence
(Note: This fraction has been adapted from mathematical work I found on the website: http://www.asahi-net.or.jp/~kc2h-msm/mathland/math05/repeat05.htm I believe the name of the mathematician who produced this website is Hisanori Mishima.
000002 000001 000003 000004 000007 000011 000018 000029 000047 000076 000123 000199 000322 000521 000843 001364 002207 003571 005778 009349 015127 024476 039603 064079 103682 167761 271443 439204 …
Terms are written in six digit strings.
Terms are accurate up to 439204, which is the 27th non-zero term.
Compare with OEIS sequence A000032.
The fraction we started with above can be rewritten as the sum of three unit fractions:
1/500000 + 1/999997000005 + 1/99999600000699999799999500000
If you take each one of these unit fractions and convert them into decimals there sum shows the Lucas Sequence. (In the final sum, the last digit is rounded up.
You might also have noticed that the denominators of the second and third fractions look like the sequence numbers for two different Fibonacci like sequences.
999,997,000,005 is the sequence number for a Fibonacci like sequence that is defined as: a(0) = 0, a(1) = 1, and when n>1 then a(n) = a(n-1) – 5*a(n-2). I would call this the 1, -5 Fibonacci sequence.
99,999,600,000,699,999,799,999,500,000 looks like a sequence number for a Fibonacci like sequence, which was then multiplied by 100,000. The Fibonacci like sequence is defined as: a(0) = a(1) = a(2) = 0, a(3) = 1, and when n>3 then a(n) = 4*a(n-1) – 7*a(n-2) + 2*a(n-3) + 4*a(n-4). I would call this the 4, -7, 2, 4 Tetranacci Sequence.
Both of these sequences produce terms that are negative. Negative terms manifest themselves in the decimal expansion in a strange way. That is why the decimal expansions shown above have the multiple occurrences of strings of nines where you would expect to see strings of zeros.
This table show the results of each of the three unit fractions used to represent the Lucas Sequence. And the fourth column shows the terms of the Lucas Sequence.
The mathematics used to combine these into a decimal expansion that shows the terms of the Lucas Sequence (written in six digit strings) can be a bit confusing when you consider that some of the terms in the table above contain more than six digits and some of the terms are negative. It's a good thing that the math knows what it is doing!
Still, it is amazing that the Lucas Sequence, which apparently cannot be expressed as one unit fraction, can be expressed as the sum of three unit fractions.
(Note: 1999999/999998999999 This fraction has been adapted from mathematical work I found on the website: http://www.asahi-net.or.jp/~kc2h-msm/mathland/math05/repeat05.htm I believe the name of the mathematician who produced this website is Hisanori Mishima. In mathematics we cannot credit all of the mathematicians who developed the mathematics that we use to produce new findings. After all, who would I credit for first finding out that one plus three equals four. But when I started this work, and needed some guidance and a hint as to which direction to proceed, I found Hisanori Mishima’s work very helpful.)