[seqfan] first post help! seeding fibonacci sequences with pythagorean triangles

Tue Jul 23 02:32:44 CEST 2019

Dear seqfans,

I hope to make my first contribution to the OEIS but I'm a bit overwhelmed
by how to go about it.

I'm only a recent member of seqfan but it already seems like most of the
math being discussed is beyond my skill level.

Despite my shortcomings, I think I have worthwhile contributions and I'd
like to try to share them with you, though you might have to be somewhat
patient with me as I try to get my points across.

Please have a glance at my questions below to see if you'd like to help.
Thank you in advance!

I'm using these formulae:

ax = c - b
a^2 + b^2 = c^2

a*(x^0) = x*(ax + 2b)
a*(x^0) = 2b*(x^1) + a*(x^2)
a*(x^2) = 2b*(x^3) + a*(x^4)
a*(x^0) = 2b*(x^1) + 2b*(x^3) + a*(x^4)
a*(x^4) = 2b*(x^5) + a*(x^6)
a*(x^0) = 2b*(x^1) + 2b*(x^3) + 2b*(x^5) + a*(x^6)
Infinite substitutions ...

f(0) = 1
f(1) = 2b/a
f(n) = f(1)*f(n-1) + f(n-2)

With these formulae, as [n] approaches Infinity, f(n-1)/f(n) approaches [x].

Since ax + b = c, f(n-1)/f(n) is also a sequence that can approximate [c]
hypotenuse lengths.

[x] defines the extreme and mean ratio (the vanishing point constant, the
metallic mean family) for all a-by-2b matrix parallelograms, not only the
pi/2 angled parallelograms (rectangles) defined by [c] hypotenuse
diagonals, and so I think it's a very important invariant (and might have
something to do with Mr. Gerdemann's recent linear recurrence multiples
question?).

Many of these f(n-1)/f(n) sequences are listed in the OEIS using "Index
entries for linear recurrences with constant coefficients",
https://oeis.org/wiki/Index_to_OEIS:_Section_Rec#order_02

a = 4, 2b = 5 is linear recurrence (5, 16) / powers of [4], which is A155455
<https://oeis.org/A155455> / A000302 <https://oeis.org/A000302> in the OEIS.

A000302(n-1) / A000302(n) = 4.

Therefore, (A155455(n-1) / A155455(n))*4*4 + 5/2 approaches [c] as [n]
approaches Infinity.

(2081/14965)*4*4 + 5/2 = 4.72492...
(780045/5630161)*4*4 + 5/2 = 4.71676...
(2158100230000361/15574988996744925)*4*4 + 5/2 = 4.71699...
(4^2 + (5/2)^2)^(1/2) = 4.71699...

Question #1:
Is worthwhile and appropriate to add the fact that A155455 approximates [c]
for the (4, 2 + (1/2), c) right triangle, as well as [x] for the 4-by-5
parallelogram, to the OEIS?

Perhaps it's better to associate A155455 with the (8, 5, c) right triangle
and the 4-by-5 parallelogram instead, preferring full reduction over
equally shared roots.

This 2b relationship between [c] and [x] parallelograms makes it difficult
to decide.

(2158100230000361/15574988996744925)*4*8 + 5 = 9.43398...
(8^2 + 5^2)^(1/2) = 9.43398...

---   ---   ---
The (3/7, 3/5, c) right triangle doesn't seem to be that uncommon.

The linear recurrence (2b, a^2) = (6/5, 9/49) that I think defines (3/7,
3/5, c)'s [x] approximating numerator seems to be much more uncommon, and
is not listed on the OEIS as far as I can tell.

(3/7, 3/5, c)'s f(n) numerator = [ 1, 14, 221, 3444, 53741, 838474,
13082161, 204112104, 3184623481 ].

(3/7, 3/5, c)'s [x] approximating f(n) denominator is defined by powers of
the fully reduced 2b/a denominator.

2b/a = (2*(3/5))/(3/7) = 14/5.

(3/7, 3/5, c)'s f(n) denominator = A000351 <https://oeis.org/A000351>.

A000351(n-1) / A000351(n) = 5.

(221/3444)*5*(3/7) + 3/5 = 0.73751...
(204112104/3184623481)*5*(3/7) + 3/5 = 0.73734...
((3/7)^2 + (3/5)^2)^(1/2) = 0.73734...

Question #2:
Is it worthwhile and appropriate to add [ 1, 14, 221, 3444, 53741, 838474,
13082161, 204112104, 3184623481 ] to the OEIS?

This sequence also approximates [c] of the (15, 21, c) right triangle and
[x] of the 15-by-42 parallelogram, which is perhaps a better way to order
it in the OEIS than as my above (3/7, 3/5, c) right triangle example.

Looking at the Theodorus Spiral as a roadmap, it seems that recursively
nesting [a] with b = 1 would index (n)^(1/2) as well as [n].

---   ---   ---
These formulae use pi/y angularity to define [y] dimensionality:

a*(x^0) = (x^(y - 1))*(ax + 2b)
a*(x^0) = 2b*(x^(y - 1)) + a*(x^y)

f(0) = 1
f(n<(y-1)) = 1
f(y-1) = 2b/a
f(n) = f(y-1)*f(n-(y-1)) + f(n-y)

a = 2, b = 3, y = 3 is linear recurrence A188022 <https://oeis.org/A188022>
.

Continuing A188022 past the OEIS list yields [
2774421135, 5216307805, 9800231959, 18423344550, 34617003682,
65070265609, 122274355596, 229827800509
].

785402143/1476968554 = 0.53177...
122274355596/229827800509 = 0.53202...
2 = (x^2)*(2x + 6); x = 0.53209...

a = 3/7, b = 3/5, y = 3, f(n) numerator = [ 1, 1, 14, 19, 221, 336, 3569,
5809, 58366, 99171, 962349, 1680224, 15952161, 28334881, 265335854,
476449139, 4423073981, 7996967216, 73834264209, 134072910929 ].

a = 3/7, b = 3/5, y = 3, f(n) denominator = A074872
<https://oeis.org/A074872>.

(336/3569)*5 = .47072...
73834264209/134072910929 = 0.55070...
(3/7) = (x^2)*((3/7)*x + 2*(3/5)); x = 0.54663...

[x] = 1/2 when a = 4, b = 7 and y = 3.

a = 4, b = 7, y = 3, f(n) numerator =
[ 1, 1, 7, 9, 53, 77, 407, 645, 3157, 5329, 24679, 43617, 194069, 354677,
1532951, 2870877, 12149365, 23162041, 96529063, 186433017, 768351605,
1498089245/1024, 6124193303, 12023327925, 48861710101, 96411682081,
390125282407, 772605194769, 3116523705173,
6188486928197, 24906086715287, 49552455907725, 199096554719797, 396679364784649
].

a = 4, b = 7, y = 3, f(n) denominator = A016116 <https://oeis.org/A016116>.

3116523705173/6188486928197 = 0.50360...
199096554719797/396679364784649 = 0.50191...
4 = (x^2)*(4x + 14); x = 0.5

Question #3
Is it worthwhile and appropriate to add any of these y = 3 dimensional
sequences to the OEIS?

The [x] = 1/2 that the [ 1, 1, 7, 9, 53, 77, 407 ] sequence approximates is
the [x] that pi/3 rotates and geometrically decays a 2-by-7 parallelogram
with pi and pi - pi/3 angles into the vanishing point, so I think it's a
relatively valuable sequence.

Question #4
I assume this f(n) recursion pattern holds true for higher values of [y],
e.g. y = 5 dimensional sequences begin with [ 1, 1, 1, 1, 2b/a ] and then
start increasing magnitude, but it'd be painstaking and error prone to
check using my plug-and-chug abilities alone. I don't know how to make a
Taylor series, for instance.

Is anyone else interested in looking into the higher values of [y] and how
they relate to higher order linear recursions? Maybe someone already did
this and I'm more ignorant than I think.

I believe order 2 linear recursions index the y = 2 sequences' numerators
in the OEIS as (2b, a^2).

---   ---   ---
Please visit https://community.wolfram.com/groups/-/m/t/1719229 for more
detail about my definitions and underlying logic, especially the
non-halting geometry-as-information dataflow thinking, based on extensively
playing with randomness infused Node.js runtime mechanics, that's driving
my research.

Sincerely,
Kyle MacLean Smith, JD
http://besta.pe