[seqfan] Decimal expansions related to Metallic Means, and Pascal Triangle

Jess Tauber yahganlang at gmail.com
Wed Aug 21 07:18:29 CEST 2013


Hi folks. I the past couple of weeks I've been looking deeper into the
structures of fractions. I started noticing certain regularities when
dealing with generalized Fibonacci sequences, and extended the search to
others where the ratio wasn't Phi but other Metallic Means.

What I found was that ALL generalized Fib sequences had decimal expansions
that were multiples of 1/89.  A021093 gives a reference to the relationship
between 1/89 and Fib, also mentioning that 120/89 is the expansion for the
Lucas numbers.

In fact, Lucas is also 19/89, which might be a more primitive answer (and
already is known, re online references). Should an addition be made to the
pages?

When I examined generalized Pell and analogous sequences which give the
Silver Mean I found that Pell had decimal expansions that were all
multiples of 1/79 (that page, A021083 has no mention of it). The next
Metallic Mean, Bronze or Copper (is there a standard here at all?), takes
1/69, the one after 1/59 and onwards subtracting 10 from the denominator
each time, though I don't know what happens when we start getting into
negatives. The pages for these expansions similarly have no mention of the
connection to the sequences or the Metallic Means they yield.

This relationship is quite systematic. Even within generalized sequences,
for example, we have rules that tell you what the next fraction should be.
In the generalized Fib sequences, if we hold the second 'seed' term
constant (so 1,1..; 2,1...; 3,1... etc.) each sequence adds 9 to the
numerator, so  1/89 for Fib, 10/89 Fib again, 19/89 for Luc, then 28, 37,
46, and so forth.

For generalized Pell numbers we add 8 to the numerator, under multiples of
1/79; the next set takes additions of 7, under 1/69, and on.

Fibonacci-like sequences in bases besides 10 are similarly regularly
motivated. Apparently for each base the basic Fib fraction has a
denominator which is a triangular number minus 1 (so for base 9, 1/71, base
10, 1/89, base 11, 1/109).

The question then arises as to whether analogues of generalized sequences
fit into this scheme, as well as those from analogues to the Metallic Means
in other bases.

Further, can we get other Pisot-Vijayaraghavan numbers into the system if
their equations are similar to those generating Metallic Means, and beyond?

On a similar note I've also been examining decimal expansions of sequences
from within the Pascal Triangle itself.   Most of you will be aware that a
summed distribution of terms from the Pascal rows, over positive powers of
ten, are powers of 11. I worked out that the edge-parallel diagonals
decimalize to negative powers of 9. The central column decimalizes to
1/sqrt60, or 1/2sqrt15, and then ratios between any contiguous pair of
columns come to a limit related to this, being 1/(5+sqrt15).

This is interesting because the decimalizations of the shallow diagonals
(which can be done in either direction), has one limit ratio (for one
direction) of 5+sqrt35. This ratio (which is in OEIS but there is no
mention of the Pascal connection) has the property that when squared you
add exactly 10 to the final result, reminiscent of the way Phi adds 1 when
squared.


Sqrt15 and sqrt35 are also of course sqrt3 and sqrt7 times sqrt5, the
latter integral to Phi, with 5=sqrt25 their constructional mean of some
sort.

Beyond mere curiosity, the reason I'm pursuing all this is that I seem to
have found an organizational link between the structure of the Pascal
system and that of the physicists M-theory. The six easiest straight-line
relations appear to parallel the six subtheories, in terms of their
dualities and connectivities, and the spacetime dimensional layouts,
alternating between extended and curled up dimensions, seems to follow
2xFib and 1xLucas sequences.



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