A391167 - OEIS

%I #29 Jul 01 2026 18:35:03

%S 26,5794,36186,111938,253786,482466,818714,1283266,1896858,2680226,

%T 3654106,4839234,6256346,7926178,9869466,12106946,14659354,17547426,

%U 20791898,24413506,28432986,32871074,37748506,43086018,48904346,55224226,62066394,69451586,77400538,85933986

%N Face-centered cubic pyritohedral numbers: a(n) = 3456*n^3 - 8424*n^2 + 6848*n - 1854.

%C These are figurate numbers produced in the construction of approximate pyritohedrons (dodecahedrons with irregular pentagonal faces) using points in a face-centered cubic lattice. The dihedral angle along the long edges for such a pyritohedron is 2*arctan(2) = A197376.

%C The n-th term is expressed as a summation which takes the (8*n-6)-th thickened cube number (A050492) and adds 6 stacks - each layer the number of points. The stacks increase by 4 layers with unit increment in n, totaling 4*n-3 layers. This corresponds to constructing a pyritohedron by taking a cube and placing a wedge on each of its 6 faces.

%C So, a(n) = A050492(8*n-6) + 6*(6*n - 4 + Sum_{i=2..n} [(6*n+2*i-8)*(8*i-14) + (6*n+2*i-9)*(8*i-13)] + [(6*n+2*i-7)*(8*i-12)+(6*n+2*i-8)*(8*i-11)] + [(6*n+2*i-8)*(8*i-10)+(6*n+2*i-7)*(8*i-9)] + [(6*n+2*i-7)*(8*i-8)+(6*n+2*i-6)*(8*i-7)]).

%H Derek Delk, <a href="/A391167/b391167.txt">Table of n, a(n) for n = 1..10000</a>

%H Derek Delk, <a href="/A391167/a391167.png">Visualization of Six Terms</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cubic_crystal_system">Cubic crystal system</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dodecahedron#Crystal_pyrite">Dodecahedron</a>.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F G.f.: 2*x*(927*x^3 + 6583*x^2 + 2845*x + 13)/(1 - x)^4.

%F E.g.f.: 2*(927 + exp(x)*(1728*x^3 + 972*x^2 + 940*x - 927)). - _Stefano Spezia_, Dec 02 2025

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

%F a(n) = a(n-1) + 10368*n^2 - 27216*n + 18728.

%F a(n) = A050492(8*n-6) + 6*[2 + Sum_{i=2..n} (704*i^2 - 1848*i + 1272)], by summing wedge gnomons.

%t a[n_]:=3456*n^3 - 8424*n^2 + 6848*n - 1854; Array[a,30] (* _James C. McMahon_, Dec 09 2025 *)

%t (* Alternative: *)

%t LinearRecurrence[{ 4,-6,4,-1},{26,5794,36186,111938},30] (* _James C. McMahon_, Dec 09 2025 *)

%t (* Alternative: *)

%t Rest[CoefficientList[Series[2*x*(927*x^3 + 6583*x^2 + 2845*x + 13)/(1 - x)^4,{x,0,30}],x]] (* _James C. McMahon_, Dec 09 2025 *)

%Y Cf. A050492, A197376.

%Y Other pyritohedral numbers: A391165, A391166.

%K nonn,easy,changed

%O 1,1

%A _Derek Delk_, Dec 01 2025