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induction_chart [2026/09/14 21:27]
paulmeyer Add link to the AI explanation for the relationship between the diagonal induction numbers and the Cotesian numbers |
induction_chart [2026/09/14 22:24] (current)
paulmeyer Improve formatting of † in the induction table |
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| |:::^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]] | | |:::^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]] | | |:::^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]] | [[m7n4|720]] | [[m8n4|45]] | [[m9n4|80]] | [[m10n4|36]] | [[m11n4|720]] † | [[m12n4|20]] | [[m13n4|720]] † | [[m14n4|180]] ? | [[m15n4|32]] | [[m16n4|45]] | [[m18n4|20]] | [[m20n4|36]] | [[m24n4|20]] | [[m30n4|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]] | [[m11n5|1440]] † | [[m12n5|20]] | [[m13n5|1440]] † | [[m14n5|180]] † | [[m15n5|32]] | [[m16n5|45]] | [[m18n5|20]] | [[m20n5|36]] | [[m24n5|20]] | [[m30n5|20]] | | |:::^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| - | - | - | - | - | - | **[[m6n6|840]]** | [[m7n6|8640]] † | [[m8n6|945]] | [[m9n6|2240]] † | [[m10n6|1512]] † | [[m11n6|60480]] † | [[m12n6|210]] | [[m13n6|60480]] † | [[m14n6|1080]] † | [[m15n6|1344]] † | [[m16n6|945]] † | [[m18n6|280]] † | [[m20n6|378]] † | [[m24n6|105]] † | [[m30n6|168]] † | | |:::^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| - | - | - | - | - | - | [[m6n7|840]] | **[[m7n7|17280]]** | [[m8n7|945]] | [[m9n7|4480]] † | [[m10n7|1512]] † | [[m11n7|120960]] † | [[m12n7|210]] † | [[m13n7|120960]] † | [[m14n7|1080]] † | [[m15n7|2688]] † | [[m16n7|945]] † | [[m18n7|280]] † | [[m20n7|378]] † | [[m24n7|105]] † | [[m30n7|168]] † | | |:::^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| - | - | - | - | - | - | - | - | **-** | [[m9n8|44800]] † | [[m10n8|9072]] † | [[m11n8|3628800]] † | [[m12n8|2100]] † | [[m13n8|3628800]] † | [[m14n8|32400]] † | [[m15n8|5376]] † | [[m16n8|14175]] † | [[m18n8|2800]] † | [[m20n8|2268]] † | [[m24n8|1050]] † | [[m30n8|336]] † | | |:::^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> | |
| |
| † 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). | <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]]. 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. |
