OFFSET
1,2
COMMENTS
These are the numbers of spheres, in the face-centered-cubic lattice packing, which form cube / octahedron intersections.
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.
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.
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
REFERENCES
Polyhedra primer / Peter Pearce and Susan Pearce. Published/Created: New York: Van Nostrand Reinhold, 1978. Description: viii, 134 pages; illustrated; 24 cm. ISBN: 0442264968
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
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n) = 48*n^3 - 108*n^2 + 80*n - 19.
G.f.: x*(1 + 89*x + 179*x^2 + 19*x^3) / (1-x)^4. - R. J. Mathar, Nov 10 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jul 01 2012
E.g.f.: (48*x^3 + 36*x^2 + 20*x - 19)*exp(x) + 19. - G. C. Greubel, Apr 28 2018
MAPLE
MATHEMATICA
CoefficientList[Series[(1+89*x+179*x^2+19*x^3)/(x-1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 01 2012 *)
PROG
(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]
(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
(GAP) List([1..50], n->(2*n-1)*(24*n^2-42*n+19)); # Muniru A Asiru, Apr 29 2018
(PARI) a(n)=(2*n-1)*(24*n^2-42*n+19) \\ Charles R Greathouse IV, Oct 18 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Chris G. Spies-Rusk (chaosorder4(AT)gmail.com), May 03 2009, May 05 2009, May 19 2009
STATUS
approved