A126562 - OEIS

%I #54 Jun 20 2024 16:58:15

%S 0,7,32,81,160,275,432,637,896,1215,1600,2057,2592,3211,3920,4725,

%T 5632,6647,7776,9025,10400,11907,13552,15341,17280,19375,21632,24057,

%U 26656,29435,32400,35557,38912,42471,46240,50225,54432,58867,63536,68445

%N Number of intersections of at least four edges in a cube of n X n X n smaller cubes.

%C a(n-1) is the number of points in a cubic lattice of n^3 equally spaced points from which all the 12*n-16 points on the 12 edges are removed. - _Luciano Ancora_, Jun 25 2015

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

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

%F G.f.: x^2*(7+4*x-5*x^2)/(1-x)^4. - _Colin Barker_, Jul 29 2012

%F a(n-1) = n^3 - (12*n-16). - _Luciano Ancora_, Jun 25 2015

%e On a cube made of 3 X 3 X 3 smaller cubes, each of the 6 sides has 4 intersections of four edges and in the center, there are 8 intersections of six edges. 6 * 4 + 8 = 32, which is a(3).

%t LinearRecurrence[{4, -6, 4, -1}, {0, 7, 32, 81}, 50] (* _Vincenzo Librandi_, Jun 27 2015 *)

%o (PARI) concat(0, Vec(x^2*(7+4*x-5*x^2)/(1-x)^4 + O(x^50))) \\ _Michel Marcus_, Jun 26 2015

%o (Magma) [6*(n-1)^2 + (n-1)^3: n in [1..40]]; // _Vincenzo Librandi_, Jun 27 2015

%o (Python) def a(n): return (n**3+3*n**2-9*n+5) # _Torlach Rush_, May 01 2024

%Y Cf. A017617.

%K nonn,easy

%O 1,2

%A Jonathan R. Love (japanada11(AT)yahoo.ca), Mar 12 2007