OFFSET
1,2
COMMENTS
These are figurate numbers produced in the construction of approximate pyritohedrons (dodecahedrons with irregular pentagonal faces) using points in a primitive cubic lattice. The dihedral angle along the long edges for such a pyritohedron is 2*arctan(2) = A197376.
The n-th term is expressed as a summation which takes (8n-7)^3 and adds 6 stacks - each layer the number of points. The stacks increase by 2 layers with unit increment in n, totaling 2n-2 layers. This corresponds to constructing a pyritohedron by taking a cube and placing a wedge on each of its 6 faces.
So, a(n) = (8*n-7)^3 + 6*Sum_{i=2..n} (6*n+2*i-9)*(8*i-15) + (6*n+2*i-9)*(8*i-11).
Using cubes (with edge lengths same as the lattice spacing) centered at these figurate points to construct such polyhedra is known as Haűy construction.
LINKS
Derek Delk, Table of n, a(n) for n = 1..10000
Derek Delk, Visualization of Six Terms
Mineralogy.eu, Wooden crystal model illustrating Haüy's laws of decrement
Eric W. Weisstein, Haűy Construction.
Wikipedia, Cubic crystal system.
Wikipedia, Dodecahedron.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
G.f.: x*(1 + 977*x + 3443*x^2 + 763*x^3)/(1 - x)^4.
E.g.f.: 763 + exp(x)*(864*x^3 + 108*x^2 + 764*x - 763). - Stefano Spezia, Dec 02 2025
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = a(n-1) + 2592*n^2 - 7560*n + 5732.
a(n) = (8*n-7)^3 + 6*Sum_{i=2..n} (176*i^2 - 556*i + 450), by summing wedge gnomons.
MATHEMATICA
LinearRecurrence[{4, -6, 4, -1}, {1, 981, 7361, 24325}, 30] (* Paolo Xausa, Dec 10 2025 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Derek Delk, Dec 01 2025
STATUS
approved