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induction_chart [2026/09/14 15:28]
paulmeyer Slightly reduce Induction table width to fit on page |
induction_chart [2026/09/14 22:24] (current)
paulmeyer Improve formatting of † in the induction table |
| /* Compact table font to fit on page */ | /* Compact table font to fit on page */ |
| .dokuwiki table.inline { font-size: 96% !important; } | .dokuwiki table.inline { font-size: 96% !important; } |
| | .dokuwiki table.inline sup { |
| | width: 0; |
| | display: inline-block; |
| | } |
| </style> | </style> |
| </HTML> | </HTML> |
| |:::^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]]. Why this is so is not clear. | **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. |
| |
| 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. | 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. |
