S-expression: Difference between revisions

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* [[#Sk*S(k + 1) (triangle-particulars)|Triangle-particulars]]: {{nowrap|'''S''k'' * S(''k'' + 1)'''}}, superparticular fractions of the form {{sfrac|''k''(''k'' + 1)/2|(''k'' − 1)(''k'' + 2)/2}}. Tempering out {{nowrap|S''k'' * S(''k'' + 1)}} equates {{sfrac|''k'' + 2|''k'' + 1}} and {{sfrac|''k''|''k'' − 1}}, or {{sfrac|''k'' + 2|''k''}} and {{sfrac|''k'' + 1|''k'' − 1}}.

* [[#Sk2 * S(k + 1) and S(k − 1) * Sk2 (lopsided commas)|Lopsided commas]]: {{nowrap|'''(S''k'')2 * S(''k'' + 1)'''}} and {{nowrap|'''(S''k'')2 * S(''k'' − 1)'''}}. Tempering out the former equates {{sfrac|''k'' + 2|''k''}} with {{pars|{{sfrac|''k''|''k'' − 1}}}}2, and tempering out the latter equates {{sfrac|''k''|''k'' − 2}} with {{pars|{{sfrac|''k'' + 1|''k''}}}}2.

* [[#Sk/S(k + 1) (ultraparticulars)|Ultraparticulars]]: {{nowrap|'''S''k''/S(''k'' + 1)'''}}. Tempering this out ~~equates~~ {{sfrac|''k'' + 2|''k'' − 1}} ~~with~~ {{pars|{{sfrac|''k'' + 1|''k''}}}}3.

* [[#Sk/S(k + 1) (ultraparticulars)|Ultraparticulars]]: {{nowrap|'''S''k''/S(''k'' + 1)'''}}. Tempering this out splits {{sfrac|''k'' + 2|''k'' − 1}} into {{pars|{{sfrac|''k'' + 1|''k''}}}}3.

* [[#Sk/S(k + 2) (semiparticulars)|Semiparticulars]]: {{nowrap|'''S''k''/S(''k'' + 2)'''}}. Tempering this out ~~equates~~ {{sfrac|''k'' + 3|''k'' − 1}} ~~with~~ {{pars|{{sfrac|''k'' + 2|''k''}}}}2.

* [[#Sk/S(k + 2) (semiparticulars)|Semiparticulars]]: {{nowrap|'''S''k''/S(''k'' + 2)'''}}. Tempering this out splits {{sfrac|''k'' + 3|''k'' − 1}} into {{pars|{{sfrac|''k'' + 2|''k''}}}}2.

== S''k'' (square-particulars) ==

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 * [[#Sk*S(k + 1) (triangle-particulars)|Triangle-particulars]]: {{nowrap|'''S''k'' * S(''k'' + 1)'''}}, superparticular fractions of the form {{sfrac|''k''(''k'' + 1)/2|(''k'' − 1)(''k'' + 2)/2}}. <br>Tempering out {{nowrap|S''k'' * S(''k'' + 1)}} equates {{sfrac|''k'' + 2|''k'' + 1}} and {{sfrac|''k''|''k'' − 1}}, or {{sfrac|''k'' + 2|''k''}} and {{sfrac|''k'' + 1|''k'' − 1}}.
 * [[#Sk2 * S(k + 1) and S(k − 1) * Sk2 (lopsided commas)|Lopsided commas]]: {{nowrap|'''(S''k'')<sup>2</sup> * S(''k'' + 1)'''}} and {{nowrap|'''(S''k'')<sup>2</sup> * S(''k'' − 1)'''}}. <br>Tempering out the former equates {{sfrac|''k'' + 2|''k''}} with {{pars|{{sfrac|''k''|''k'' − 1}}}}<sup>2</sup>, and tempering out the latter equates {{sfrac|''k''|''k'' − 2}} with {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>2</sup>.
-* [[#Sk/S(k + 1) (ultraparticulars)|Ultraparticulars]]: {{nowrap|'''S''k''/S(''k'' + 1)'''}}. Tempering this out equates {{sfrac|''k'' + 2|''k'' − 1}} with {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>3</sup>.
+* [[#Sk/S(k + 1) (ultraparticulars)|Ultraparticulars]]: {{nowrap|'''S''k''/S(''k'' + 1)'''}}. Tempering this out splits {{sfrac|''k'' + 2|''k'' − 1}} into {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>3</sup>.
-* [[#Sk/S(k + 2) (semiparticulars)|Semiparticulars]]: {{nowrap|'''S''k''/S(''k'' + 2)'''}}. Tempering this out equates {{sfrac|''k'' + 3|''k'' − 1}} with {{pars|{{sfrac|''k'' + 2|''k''}}}}<sup>2</sup>.
+* [[#Sk/S(k + 2) (semiparticulars)|Semiparticulars]]: {{nowrap|'''S''k''/S(''k'' + 2)'''}}. Tempering this out splits {{sfrac|''k'' + 3|''k'' − 1}} into {{pars|{{sfrac|''k'' + 2|''k''}}}}<sup>2</sup>.
 == S''k'' (square-particulars) ==