Differences

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--- induction_chart [2023/08/08 15:06]
paulmeyer Update the induction chart with entry for n=4, m=12
+++ induction_chart [2026/09/14 22:24] (current)
paulmeyer Improve formatting of † in the induction table
@@ Line 1: / Line 1: @@
+<HTML>
+<style>
+/* Compact table font to fit on page */
+.dokuwiki table.inline { font-size: 96% !important; }
+.dokuwiki table.inline sup {
+    width: 0;
+    display: inline-block;
+}
+</style>
+</HTML>
 ====Induction Chart====
 | |^  //m// copies of the original //n//-die set  |||||||||||||||||||||
@@ Line 6: / Line 17: @@
 |:::^2|  -  |  -  |  **[[m2n2|6]]**  |  [[m3n2|4]]  |  [[m4n2|6]]  |  [[m5n2|12]]  |  [[m6n2|4]]  |  [[m7n2|12]]  |  [[m8n2|6]]  |  [[m9n2|4]]  |  [[m10n2|6]]  |  [[m11n2|12]]  |  [[m12n2|4]]  |  [[m13n2|12]]  |  [[m14n2|6]]  |  [[m15n2|4]]  |  [[m16n2|6]]  |  [[m18n2|4]]  |  [[m20n2|6]]  |  [[m24n2|4]]  |  [[m30n2|4]]  |
 |:::^3|  -  |  -  |  [[m2n3|6]]  |  **[[m3n3|8]]**  |  [[m4n3|6]]  |  [[m5n3|24]]  |  [[m6n3|6]]  |  [[m7n3|24]]  |  [[m8n3|6]]  |  [[m9n3|8]]  |  [[m10n3|6]]  |  [[m11n3|24]]  |  [[m12n3|6]]  |  [[m13n3|24]]  |  [[m14n3|6]]  |  [[m15n3|8]]  |  [[m16n3|6]]  |  [[m18n3|6]]  |  [[m20n3|6]]  |  [[m24n3|6]]  |  [[m30n3|6]]  |
-|:::^4|  -  |  -  |  -  |  -  |  **[[m4n4|90]]**  |  [[m5n4|144]]  |  [[m6n4|20]]  |  [[m7n7|720]] ?  |  [[m8n4|45]]  |  [[m9n4|80]]  |  [[m10n4|36]]  |  |  [[m12n4|20]]  |  |  |  |  |  |  |  |  20  |
+|:::^4|  -  |  -  |  -  |  -  |  **[[m4n4|90]]**  |  [[m5n4|144]]  |  [[m6n4|20]]  |  [[m7n4|720]]  |  [[m8n4|45]]  |  [[m9n4|80]]  |  [[m10n4|36]]  |  [[m11n4|720]]<sup>†</sup>  |  [[m12n4|20]]  |  [[m13n4|720]]<sup>†</sup>  |  [[m14n4|180]] ?  |  [[m15n4|32]]  |  [[m16n4|45]]  |  [[m18n4|20]]  |  [[m20n4|36]]  |  [[m24n4|20]]  |  [[m30n4|20]]  |
-|:::^5|  -  |  -  |  -  |  -  |  [[m4n5|90]]  |  **[[m5n5|288]]**  |  [[m6n5|20]]  |  [[m7n5|1440]]  |  [[m8n5|45]]  |  [[m9n5|160]]  |  [[m10n5|36]]  |  |  |  |  |  |  |  |  |  |  |
+|:::^5|  -  |  -  |  -  |  -  |  [[m4n5|90]]  |  **[[m5n5|288]]**  |  [[m6n5|20]]  |  [[m7n5|1440]]  |  [[m8n5|45]]  |  [[m9n5|160]]  |  [[m10n5|36]]  |  [[m11n5|1440]]<sup>†</sup>  |  [[m12n5|20]]  |  [[m13n5|1440]]<sup>†</sup>  |  [[m14n5|180]]<sup>†</sup>  |  [[m15n5|32]]  |  [[m16n5|45]]  |  [[m18n5|20]]  |  [[m20n5|36]]  |  [[m24n5|20]]  |  [[m30n5|20]]  |
-|:::^6|  -  |  -  |  -  |  -  |  -  |  -  |  **840**  |  8640 ?  |  945  |  ?  |  ?  |  |  |  |  |  |  |  |  |  |  |
+|:::^6|  -  |  -  |  -  |  -  |  -  |  -  |  **[[m6n6|840]]**  |  [[m7n6|8640]]<sup>†</sup>  |  [[m8n6|945]]  |  [[m9n6|2240]]<sup>†</sup>  |  [[m10n6|1512]]<sup>†</sup>  |  [[m11n6|60480]]<sup>†</sup>  |  [[m12n6|210]]  |  [[m13n6|60480]]<sup>†</sup>  |  [[m14n6|1080]]<sup>†</sup>  |  [[m15n6|1344]]<sup>†</sup>  |  [[m16n6|945]]<sup>†</sup>  |  [[m18n6|280]]<sup>†</sup>  |  [[m20n6|378]]<sup>†</sup>  |  [[m24n6|105]]<sup>†</sup>  |  [[m30n6|168]]<sup>†</sup>  |
-|:::^7|  -  |  -  |  -  |  -  |  -  |  -  |  840  |  **17280 ?**  |  945  |  ?  |  ?  |  |  |  |  |  |  |  |  |  |  |
+|:::^7|  -  |  -  |  -  |  -  |  -  |  -  |  [[m6n7|840]]  |  **[[m7n7|17280]]**  |  [[m8n7|945]]  |  [[m9n7|4480]]<sup>†</sup>  |  [[m10n7|1512]]<sup>†</sup>  |  [[m11n7|120960]]<sup>†</sup>  |  [[m12n7|210]]<sup>†</sup>  |  [[m13n7|120960]]<sup>†</sup>  |  [[m14n7|1080]]<sup>†</sup>  |  [[m15n7|2688]]<sup>†</sup>  |  [[m16n7|945]]<sup>†</sup>  |  [[m18n7|280]]<sup>†</sup>  |  [[m20n7|378]]<sup>†</sup>  |  [[m24n7|105]]<sup>†</sup>  |  [[m30n7|168]]<sup>†</sup>  |
-|:::^8|  -  |  -  |  -  |  -  |  -  |  -  |  -  |  -  |  ?  |  ?  |  ?  |  |  |  |  |  |  |  |  |  |  |
+|:::^8|  -  |  -  |  -  |  -  |  -  |  -  |  -  |  -  |  **-**  |  [[m9n8|44800]]<sup>†</sup>  |  [[m10n8|9072]]<sup>†</sup>  |  [[m11n8|3628800]]<sup>†</sup>  |  [[m12n8|2100]]<sup>†</sup>  |  [[m13n8|3628800]]<sup>†</sup>  |  [[m14n8|32400]]<sup>†</sup>  |  [[m15n8|5376]]<sup>†</sup>  |  [[m16n8|14175]]<sup>†</sup>  |  [[m18n8|2800]]<sup>†</sup>  |  [[m20n8|2268]]<sup>†</sup>  |  [[m24n8|1050]]<sup>†</sup>  |  [[m30n8|336]]<sup>†</sup>  |
+<sup>†</sup> These entries were calculated with the help of AI (Gemini), and may need further manual checking to verify minimality. The values themselves and all linked schematics have already been manually checked to be valid (just not necessarily minimal).
+**Bold** numbers highlight that those numbers along the diagonal match the sums of the rows of the [[http://oeis.org/A100642/table|triangular array of Cotesian Numbers]]. The reason for this was not clear until recently (Sept 10, 2026), when an AI (Gemini) was used to generate [[induction_and_cotesian_numbers|this mathematical explanation]]. In summary, the reason is because the induction equations reduce to be algebraically identical to the Newton-Cotes quadrature equations when //n//=//m//, of which the Cotesian numbers are the explicit and unique solutions.
-Bold numbers highlight that those numbers along the diagonal match the sums of the rows of the [[http://oeis.org/A100642/table|triangular array of Cotesian Numbers]]. Why this is so is not clear.
+Another interesting pattern is that the columns where //m// is prime appear to follow the [[https://oeis.org/A091137|Hirzebruch numbers]] for all //n//<(//m-1//), and then instead follow //1/m//<sup>th</sup> of the remaining Hirzebruch numbers.
 ===How This Chart Can Be Used===

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