A160174 a(n) = (2*n - 1)*(24*n^2 - 42*n + 19).

1, 93, 545, 1645, 3681, 6941, 11713, 18285, 26945, 37981, 51681, 68333, 88225, 111645, 138881, 170221, 205953, 246365, 291745, 342381, 398561, 460573, 528705, 603245, 684481, 772701, 868193, 971245, 1082145, 1201181, 1328641, 1464813, 1609985, 1764445

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

A160174:=n->(2*n-1)*(24*n^2-42*n+19); seq(A160174(n), n=1..40); # Wesley Ivan Hurt, Jun 21 2014

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

Sequence in context: A116240 A193248 A146090 * A238693 A160250 A332614

Adjacent sequences: A160171 A160172 A160173 * A160175 A160176 A160177

Keyword

nonn,easy

Author

Chris G. Spies-Rusk (chaosorder4(AT)gmail.com), May 03 2009, May 05 2009, May 19 2009

Status

approved