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Metallic numbers: Beyond the golden ratio Metallic numbers: Fibonacci and more

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Sergio

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Jan 19, 2021, 1:01:31 PM1/19/21
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https://plus.maths.org/content/silver-ratio

https://plus.maths.org/content/part-ii

djoyce099

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Jan 19, 2021, 1:47:14 PM1/19/21
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On Tuesday, January 19, 2021 at 1:01:31 PM UTC-5, Sergio wrote:
> https://plus.maths.org/content/silver-ratio
>
> https://plus.maths.org/content/part-ii

Just starting these sequences with their starting number like the silver mean as
sqrt(2) +1*2 =4.82842...rounded up =5
2,5,
sqrt(2) +1*5 =12.07106... rounded down =12
2,5,12
sqrt(2) +1*12 =28.97056... rounded up =29
2,5,12,29
And soforth for all the remaining metallic mean sequences to present my argument.

So yes, a few starting integers in each sequence are missing such as 0 and 1.
So that produces a column of starting integers as ---
1..
2..
3..
4..
..
That column is then eliminated to not produce a duplicate except the 5 just to
present my argument of no more duplicates other than the two 5's

djoyce099

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Jan 19, 2021, 2:14:25 PM1/19/21
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Checking with OEIS these leading 0's and 1's are not consistent with these sequences.
Some show both 0 and 1 and some only show 1.
So starting with the first column, as shown above, eliminates any confusion about this matrix
That I presented in my original post.


djoyce099

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Jan 19, 2021, 2:17:30 PM1/19/21
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That also produces the second column of the first column of n as n^2+1
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