1,2

From Derek Delk, Oct 15 2025: (Start)

These are figurate numbers produced in the construction of approximate rhombic dodecahedrons using points in a face-centered cubic lattice. Each term is expressed as two summations.

One summation takes the (4n-3)-th octahedral number (A005900) and adds 8 stacks of triangular numbers (A000217), each consisting of n-1 layers. This corresponds to constructing a rhombic dodecahedron by starting with an octahedron and placing a triangular pyramid on each of its 8 faces.

So, a(n) = A005900(4n-3) + 8*Sum_{i=1..n-1} A000217(4i-3).

Another summation takes the (2n-1)-th thickened cube number (A050492) and adds 6 stacks consisting of alternating layers of centered square numbers (A001844) and 4 times triangular numbers (A046092), totaling 2n-2 layers per stack. This corresponds to constructing a rhombic dodecahedron by starting with a cube and placing a square pyramid on each of its 6 faces.

So, a(n) = A050492(2n-1) + 6*Sum_{i=2..n} A001844(2i-4) + A046092(2i-3).

These figurate numbers relate to face-centered cubic lattice analogously to how rhombic dodecahedral numbers (A005917) relate to body-centered cubic lattice and how Haüy rhombic dodecahedral numbers (A046142) relate to primitive cubic lattice. (End)

John H. Conway and Richard K. Guy, 'The Book of Numbers', Copernicus Pub. Co., 1996.

Peter Pearce and Susan Pearce, 'Polyhedra primer', Van Nostrand Reinhold, 1978.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Derek Delk, Visualization of Six Terms

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

G.f.: x*(1 + 89*x + 251*x^2 + 43*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.: 43 + (-43 + 44*x + 24*x^2 + 64*x^3)*exp(x). - G. C. Greubel, Nov 14 2019

seq(((8*n-7)^3 +(8*n-1))/8, n=1..40); # G. C. Greubel, Nov 14 2019

CoefficientList[Series[(1+89*x+251*x^2+43*x^3)/(x-1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 01 2012 *)

LinearRecurrence[{4, -6, 4, -1}, {1, 93, 617, 1957}, 30] (* Harvey P. Dale, Mar 16 2013 *)

Table[((8*n-7)^3 +(8*n-1))/8, {n, 40}] (* G. C. Greubel, Nov 14 2019 *)

(Magma) I:=[1, 93, 617, 1957]; [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

(PARI) a(n)=64*n^3-168*n^2+148*n-43 \\ Charles R Greathouse IV, Apr 25 2016

(SageMath) [((8*n-7)^3 +(8*n-1))/8 for n in (1..40)] # G. C. Greubel, Nov 14 2019

(GAP) List([1..40], n-> ((8*n-7)^3 +(8*n-1))/8); # G. C. Greubel, Nov 14 2019

Sequence in context: A146090 A160174 A238693 * A332614 A264556 A250373

Adjacent sequences: A160247 A160248 A160249 * A160251 A160252 A160253

easy,nonn

Chris G. Spies-Rusk (chaosorder4(AT)gmail.com), May 05 2009, May 11 2009, May 19 2009

Edited by G. C. Greubel, Nov 14 2019

approved