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%I #37 Oct 18 2022 15:17:16
%S 1,93,545,1645,3681,6941,11713,18285,26945,37981,51681,68333,88225,
%T 111645,138881,170221,205953,246365,291745,342381,398561,460573,
%U 528705,603245,684481,772701,868193,971245,1082145,1201181,1328641,1464813,1609985,1764445
%N a(n) = (2*n - 1)*(24*n^2 - 42*n + 19).
%C These are the numbers of spheres, in the face-centered-cubic lattice packing, which form cube / octahedron intersections.
%C The edges of the respective hedra intersect perpendicularly at their midpoints. The edges for the octahedra are series of contiguous units, numbering 4n-3. The edges for the cubes are intermittent series of units numbering 2n-1. The gap between spheres is root2-1.
%C The vertices of this non-convex form are the same as those of the convex rhombic dodecahedra in A160250. That one results from "shrink wrapping" this one.
%C These numbers are never prime because the polynomial factors over Z into a(n) = (2*n-1)*(24*n^2 - 42*n + 19). It is semiprime when both factors are prime, as for 93 = 3 * 31, 545 = 5 * 109, 6941 = 11 * 631, 37981 = 19 * 1999, 68333 = 23 * 2971, 138881 = 29 * 4789, 398561 = 41 * 9721, 460573 = 43 * 10711, 868193 = 53 * 16381, 1201181 = 59 * 20359. - _Jonathan Vos Post_ Dec 15 2010
%D Polyhedra primer / Peter Pearce and Susan Pearce. Published/Created: New York: Van Nostrand Reinhold, 1978. Description: viii, 134 pages; illustrated; 24 cm. ISBN: 0442264968
%D The book of numbers / John H. Conway, Richard K. Guy. Published/Created: New York, NY: Copernicus, 1996. Description: ix, 310 pages; illustrated (some color); 24 cm. ISBN: 038797993X
%H Vincenzo Librandi, <a href="/A160174/b160174.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = 48*n^3 - 108*n^2 + 80*n - 19.
%F G.f.: x*(1 + 89*x + 179*x^2 + 19*x^3) / (1-x)^4. - _R. J. Mathar_, Nov 10 2011
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jul 01 2012
%F E.g.f.: (48*x^3 + 36*x^2 + 20*x - 19)*exp(x) + 19. - _G. C. Greubel_, Apr 28 2018
%p A160174:=n->(2*n-1)*(24*n^2-42*n+19); seq(A160174(n), n=1..40); # _Wesley Ivan Hurt_, Jun 21 2014
%t CoefficientList[Series[(1+89*x+179*x^2+19*x^3)/(x-1)^4,{x,0,40}],x] (* _Vincenzo Librandi_, Jul 01 2012 *)
%o (Excel) The following formula will give volumes corresponding to row numbers as n when filled down in a column. =48*ROW()^3-108*ROW()^2+80*ROW()-19 [From Chris G. Spies-Rusk (chaosorder4(AT)gmail.com), May 05 2009]
%o (Magma) I:=[1, 93, 545, 1645]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Jul 01 2012
%o (GAP) List([1..50],n->(2*n-1)*(24*n^2-42*n+19)); # _Muniru A Asiru_, Apr 29 2018
%o (PARI) a(n)=2*n-1)*(24*n^2-42*n+19 \\ _Charles R Greathouse IV_, Oct 18 2022
%K nonn,easy
%O 1,2
%A Chris G. Spies-Rusk (chaosorder4(AT)gmail.com), May 03 2009, May 05 2009, May 19 2009
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