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- Pentagonal_number abstract "A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside.pn is given by the formula:for n ≥ 1. The first few pentagonal numbers are:1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001 (sequence A000326 in OEIS).The nth pentagonal number is one third of the 3n-1th triangular number.Generalized pentagonal numbers are obtained from the formula given above, but with n taking values in the sequence 0, 1, -1, 2, -2, 3, -3, 4..., producing the sequence:0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335... (sequence A001318 in OEIS).Generalized pentagonal numbers are important to Euler's theory of partitions, as expressed in his pentagonal number theorem.The number of dots inside the outermost pentagon of a pattern forming a pentagonal number is itself a generalized pentagonal number.Pentagonal numbers should not be confused with centered pentagonal numbers.".
- Pentagonal_number thumbnail Pentagonal_number.gif?width=300.
- Pentagonal_number wikiPageExternalLink 0505373.
- Pentagonal_number wikiPageID "504786".
- Pentagonal_number wikiPageRevisionID "604536977".
- Pentagonal_number hasPhotoCollection Pentagonal_number.
- Pentagonal_number subject Category:Figurate_numbers.
- Pentagonal_number type Abstraction100002137.
- Pentagonal_number type Amount105107765.
- Pentagonal_number type Attribute100024264.
- Pentagonal_number type FigurateNumbers.
- Pentagonal_number type Magnitude105090441.
- Pentagonal_number type Number105121418.
- Pentagonal_number type Property104916342.
- Pentagonal_number comment "A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex.".
- Pentagonal_number label "Fünfeckszahl".
- Pentagonal_number label "Nombre pentagonal".
- Pentagonal_number label "Numero pentagonale".
- Pentagonal_number label "Número pentagonal".
- Pentagonal_number label "Pentagonal number".
- Pentagonal_number label "Vijfhoeksgetal".
- Pentagonal_number label "عدد مخمسي".
- Pentagonal_number label "五角数".
- Pentagonal_number label "五角数".
- Pentagonal_number sameAs Pětiúhelníkové_číslo.
- Pentagonal_number sameAs Fünfeckszahl.
- Pentagonal_number sameAs Número_pentagonal.
- Pentagonal_number sameAs Nombre_pentagonal.
- Pentagonal_number sameAs Numero_pentagonale.
- Pentagonal_number sameAs 五角数.
- Pentagonal_number sameAs 오각수.
- Pentagonal_number sameAs Vijfhoeksgetal.
- Pentagonal_number sameAs m.02jf0r.
- Pentagonal_number sameAs Q674928.
- Pentagonal_number sameAs Q674928.
- Pentagonal_number sameAs Pentagonal_number.
- Pentagonal_number wasDerivedFrom Pentagonal_number?oldid=604536977.
- Pentagonal_number depiction Pentagonal_number.gif.
- Pentagonal_number isPrimaryTopicOf Pentagonal_number.
