S-expression: Difference between revisions

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== S(k - 1)*Sk*S(k + 1) (1/3-square-particulars) ==

This section concerns commas of the form S(''k'' - 1) * S''k'' * S(''k'' + 1) = ( (k-1)/(k-2) )/( (k+2)/(k+1) ) which therefore do not (directly) involve the ''k''th harmonic.

=== Significance ===

1. Two thirds of all 1/3-square-particulars are superparticular and the other third are [[#Glossary|throdd-particular]], so these are efficient commas. (See also the [[#Proof of simplification of 1/3-square-particulars]].)

2. They are often implied in a variety of ways by combinations of other commas discussed on this page.

3. Their omission of direct relation to the ''k''th harmonic make them theoretically interesting and potentially useful. (The other type of comma on this page that does this is [[semiparticular]]s.)

4. Square-particulars, 1/2-square-particulars (a.k.a. [[triangle-particular]]s) and 1/3-square-particulars are part of a more general sequence with interesting properties: [[1/n-square-particular]]s.

=== Proof of simplification of 1/3-square-particulars ===

This section concerns commas of the form S(''k'' - 1) * S''k'' * S(''k'' + 1) = ( (k-1)/(k-2) )/( (k+2)/(k+1) ) which therefore do not (directly) involve the kth harmonic. We can check their general algebraic expression for any potential simplifications:

<pre>

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In other words, what this shows is all 1/3-square-particulars of the form S(''k'' - 1) * S''k'' * S(''k'' + 1) are superparticular iff ''k'' is throdd (not a multiple of 3), and all 1/3-square-particulars of the form S(3''k'' - 1) * S(3''k'') * S(3''k'' + 1) are throdd-particular with the numerator and denominator always being one less than a multiple of 3 (which is to say, commas of this form are throdd-particular iff k is threven and superparticular iff k is throdd).

=== Table of 1/3-square-particulars ===

Below is a table of such commas in the 41-prime-limited 199-odd-limit:

{| class="wikitable center-all

@@ Line 391: / Line 391: @@
 == S(k - 1)*Sk*S(k + 1) (1/3-square-particulars) ==
+This section concerns commas of the form S(''k'' - 1) * S''k'' * S(''k'' + 1) = ( (k-1)/(k-2) )/( (k+2)/(k+1) ) which therefore do not (directly) involve the ''k''th harmonic.
+=== Significance ===
+. Two thirds of all 1/3-square-particulars are superparticular and the other third are [[#Glossary|throdd-particular]], so these are efficient commas. (See also the [[#Proof of simplification of 1/3-square-particulars]].)
+. They are often implied in a variety of ways by combinations of other commas discussed on this page.
+. Their omission of direct relation to the ''k''th harmonic make them theoretically interesting and potentially useful. (The other type of comma on this page that does this is [[semiparticular]]s.)
+. Square-particulars, 1/2-square-particulars (a.k.a. [[triangle-particular]]s) and 1/3-square-particulars are part of a more general sequence with interesting properties: [[1/n-square-particular]]s.
+=== Proof of simplification of 1/3-square-particulars ===
 This section concerns commas of the form S(''k'' - 1) * S''k'' * S(''k'' + 1) = ( (k-1)/(k-2) )/( (k+2)/(k+1) ) which therefore do not (directly) involve the kth harmonic. We can check their general algebraic expression for any potential simplifications:
 <pre>
@@ Line 405: / Line 417: @@
 In other words, what this shows is all 1/3-square-particulars of the form S(''k'' - 1) * S''k'' * S(''k'' + 1) are superparticular iff ''k'' is throdd (not a multiple of 3), and all 1/3-square-particulars of the form S(3''k'' - 1) * S(3''k'') * S(3''k'' + 1) are throdd-particular with the numerator and denominator always being one less than a multiple of 3 (which is to say, commas of this form are throdd-particular iff k is threven and superparticular iff k is throdd).
+=== Table of 1/3-square-particulars ===
 Below is a table of such commas in the 41-prime-limited 199-odd-limit:
 {| class="wikitable center-all