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Nombre octaédrique

Un nombre octaédrique est un nombre figuré polyédrique comptant des points régulièrement répartis dans un octaèdre régulier, ou deux pyramides placées ensemble, l'une placée sur l'autre renversée.

Le nombre octaédrique d'ordre $n$ , correspondant au cas où il y a $n$ points sur chaque arête de l'octaédre, est donné par la formule :

$O_{n}={n(2n^{2}+1) \over 3}.$ ^[1]^,^[2]^,^[3].

Les dix premiers nombres octaédriques sont :

1, 6, 19, 44, 85, 146, 231, 344, 489, 670 (suite A005900 de l'OEIS).

La série génératrice des nombres octaédriques est la fraction rationnelle :

{\frac {z(z+1)^{2}}{(z-1)^{4}}}=\sum _{n=1}^{\infty }O_{n}z^{n}=z+6z^{2}+19z^{3}+\cdots .

Obtention du nombre octaédrique d'ordre n

Comme somme de deux nombres pyramidaux

O_n peut être obtenu en ajoutant deux nombres pyramidaux carrés consécutifs : $O_{n}=\sum _{k=1}^{n-1}k^{2}+\sum _{k=1}^{n}k^{2}={\frac {(n-1)n(2n-1)}{6}}+{\frac {n(n+1)(2n+1)}{6}}={n(2n^{2}+1) \over 3}.$

Par la construction générale des nombres polyédriques réguliers

Article détaillé : Nombre polyédrique.

On obtient ici $O_{n}$ à partir de la relation : $O_{n}-O_{n-1}=(S-1)+(A-q)(n-2)+(F-q)(P_{m,n}-m(n-1))$ ,

où $S=6,A=12,F=8$ sont les nombres de sommets, arêtes et faces de l'octaèdre, $\{m,q\}=\{3,4\}$ son symbole de Schläfli : {nombre d'arêtes par face, nombre d'arêtes (et aussi de faces) par sommet} , et $P_{m,n}$ le nombre m-gonal d'ordre n ^[2].

On obtient donc $O_{n}-O_{n-1}=(6-1)+(12-4)(n-2)+(8-4)(n(n+1)/2-3(n-1))=2n^{2}-2n+1$ .

D'où $O_{n}=2{\frac {n(n+1)(2n+1)}{6}}-2{\frac {n(n+1)}{2}}+n={n(2n^{2}+1) \over 3}$ .

Nombre octaédrique tronqué

Si l'on retranche à chacun des 6 sommets de la construction précédente à l'étape $3n-2$ une pyramide à base carrée à l'étape $n-1$ , on obtient les nombres octaédriques tronqués : $OT_{n}=O_{3n-2}-6P_{n-1}^{(4)}=16n^{3}-33n^{2}+24n-6$ : 1, 38, 201, 586, 1289, 2406, 4033, 6266, 9201, 12934,... (suite A005910 de l'OEIS) ^[4].

Références

↑ (en) Hyun Kwang Kim, « On Regular Polytope Numbers », PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, vol. 131, n^o 1,‎ 2002, p. 68 (lire en ligne)
↑ ^{a et b} (en) Elena Deza et Michel Deza, Figurate Numbers, Singapour, World Scientific Publishing, 2012, 456 p. (ISBN 978-981-4355-48-3, lire en ligne), p. 105-109
↑ Charles-É. Jean, « Nombre octaédrique ou octaédrique D3 », sur Récréomath
↑ John H. Conway, Richard K.Guy, Le livre des nombres, Eyrolles, 1998, p. 52-53

(en) Cet article est partiellement ou en totalité issu de l’article de Wikipédia en anglais intitulé « Octahedral number » (voir la liste des auteurs).

Articles connexes

Lien externe

(en) Eric W. Weisstein, « Octahedral number », sur MathWorld

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Nombres figurés

Bidimensionnel

Polygonal non centré	Triangulaire Carré Triangulaire carré Trapézoïdal Pentagonal Hexagonal Heptagonal Octogonal Ennéagonal Décagonal Dodécagonal
Polygonal centré	Triangulaire centré Carré centré Pentagonal centré Hexagonal centré Heptagonal centré Octogonal centré Ennéagonal centré Décagonal centré Etoilé

Tridimensionnel

Polyédrique non centré	Tétraédrique Cubique Octaédrique Dodécaédrique Icosaédrique Octaédrique tronqué Pyramidal
Polyédrique centré	Tétraédrique centré Cubique centré Octaédrique centré Dodécaédrique centré Icosaédrique centré
Pyramidal	Tétraédrique Carré Pentagonal Hexagonal Heptagonal

Quadridimensionnel

4-polytopique non centré	Pentatopique (hypertétraèdrique) 4-hypercubique
4-polytopique centré	Pentatopique centré

Multidimensionnel

Polytopique non centré
Polytopique centré

Arithmétique et théorie des nombres

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