A103661 -id:A103661

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A103157

Number of ways to choose 4 distinct points from an (n+1) X (n+1) X (n+1) lattice cube.

+10
4

70, 17550, 635376, 9691375, 88201170, 566685735, 2829877120, 11671285626, 41417124750, 130179173740, 370215608400, 968104633665, 2357084537626, 5396491792125, 11710951848960, 24246290643940, 48151733324310, 92140804597626, 170538695998000, 306294282269955

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

FORMULA

a(n) = binomial((n+1)^3, 4).

G.f.: -x*(x^10 + 317*x^9 + 23193*x^8 + 435669*x^7 + 2747685*x^6 + 6738399*x^5 + 6803373*x^4 + 2780367*x^3 + 412686*x^2 + 16640*x + 70)/(x -1)^13. - Colin Barker, Nov 16 2012

CROSSREFS

Cf. 4-point objects in lattice cube: A103158 tetrahedra, A103656 triangular pyramids, A103657 number of different volumes, A103658 volume=0, A103659, A103660 most frequent volumes, A103661 smallest not occurring volume.

KEYWORD

easy,nonn

AUTHOR

Hugo Pfoertner, Feb 12 2005

STATUS

approved

A103657

Number of different volumes assumed by triangular pyramids with their 4 vertices chosen from distinct points of an (n+1)X(n+1)X(n+1) lattice cube, including degenerate objects with volume=0.

+10
4

3, 13, 39, 90, 178, 309, 503, 756, 1096, 1523, 2059, 2683, 3469, 4355, 5406

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..15.

EXAMPLE

a(1)=3 because 4-point objects with 3 different volumes can be built using the vertices of a cube: 2 regular tetrahedra (e.g. [(0,0,0),(0,1,1),(1,0,1),(1,1,0)]) with volume 1/3, 56 pyramids with volume 1/6 and 12 objects with volume=0, e.g. the faces of the cube.

a(2)=13: The A103157(2)=17550 4-point objects that can selected from the 27 points of a 3X3X3 lattice cube fall into 13 different volume classes (6*V,occurrences):

(0,2918), (1,3688), (2,5272), (3,1272), (4,2788), (5,272), (6,684), (7,72), (8,494), (9,16), (10,48), (12,24), (16,2).

A103658(n) gives the occurrence counts of objects with V=0 (i.e. A103658(2)=2918).

A103659(n) gives 6*V of the most frequently occurring volume and A103660(n) gives the corresponding occurrence count, divided by 2. Therefore A103659(2)=2 and A103660(2)=2636.

A103661(n) gives the smallest value of 6*V not occurring in the list of 4-point object volumes, i.e. A103661(2)=11.

CROSSREFS

Cf. A103157 binomial((n+1)^3, 4), A103158 tetrahedra in lattice cube, A103656, A103658, A103659, A103660, A103661.

KEYWORD

hard,nonn

AUTHOR

Hugo Pfoertner, Feb 17 2005

STATUS

approved

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