A108099 a(n) = 8*n^2 + 8*n + 4.

4, 20, 52, 100, 164, 244, 340, 452, 580, 724, 884, 1060, 1252, 1460, 1684, 1924, 2180, 2452, 2740, 3044, 3364, 3700, 4052, 4420, 4804, 5204, 5620, 6052, 6500, 6964, 7444, 7940, 8452, 8980, 9524, 10084, 10660, 11252, 11860, 12484, 13124, 13780, 14452, 15140, 15844

Offset: 0

Author

Dorthe Roel (dorthe_roel(AT)hotmail.com or dorthe.roel1(AT)skolekom.dk), Jun 07 2005

Keywords

• nonn
• easy

Comments

Also the number for Waterman [polyhedra] have a unit rhombic dodecahedron face so sqrt 4, sqrt 20, sqrt 52, etc...and a one-to-one match...that is, no omissions and no extras. - Steve Waterman and Roger Kaufman (swaterman(AT)watermanpolyhedron.com), Apr 02 2009. [This sentence makes no sense - some words must have been dropped. - N. J. A. Sloane, Jun 12 2014]

Also, sequence found by reading the segment (4,20) together with the line from 20, in the direction 20, 52, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Sep 04 2011

Sum of consecutive even squares: (2*n)^2 + (2*n + 2)^2 = 8*n^2 + 8*n + 4. - Michel Marcus, Jan 27 2014

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• Adrian Rossiter, Antiprism.
• Steve Waterman, Polyhedra Project.
• Steve Waterman, Waterman's Polyhedral Mensuration chart.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217, A001844, A016742, A035008, A069129, A069894.

Programs

Magma

[ 8*n^2 + 8*n + 4 : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

Maple

A108099:=n->8*n^2 + 8*n + 4; seq(A108099(n), n=0..50); # Wesley Ivan Hurt, Jun 09 2014

Mathematica

CoefficientList[Series[-(4*(z^2 + 2*z + 1))/(z - 1)^3, {z, 0, 100}], z] (* and *) Table[8*n*(n + 1) + 4, {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 17 2011 *)

PARI

a(n)=8*n^2+8*n+4 \\ Charles R Greathouse IV, Jul 17 2011

Formula

a(n) = 8*n^2 + 8*n + 4.

G.f.: 4*(1+2*x+x^2)/(1-x)^3.

a(n) = 16*n + a(n-1), a(0)=4. - Vincenzo Librandi, Nov 13 2010

a(n) = A069129(n+1) + 3. - Omar E. Pol, Sep 04 2011

a(n) = A035008(n) + 4. - Omar E. Pol, Jun 12 2014

From Elmo R. Oliveira, Oct 27 2024: (Start)

E.g.f.: 4*(1 + 4*x + 2*x^2)*exp(x).

a(n) = 4*A001844(n) = 2*A069894(n).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)