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

%I

%S 4,20,52,100,164,244,340,452,580,724,884,1060,1252,1460,1684,1924,

%T 2180,2452,2740,3044,3364,3700,4052,4420,4804,5204,5620,6052,6500,

%U 6964,7444,7940,8452,8980,9524,10084,10660,11252,11860,12484,13124,13780,14452,15140

%N a(n) = 8n^2 + 8n + 4.

%C 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]

%C 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

%C Sum of consecutive even squares: (2*n)^2 + (2*n+2)^2 = 8*n^2 + 8*n + 4. - _Michel Marcus_, Jan 27 2014

%H Ivan Panchenko, <a href="/A108099/b108099.txt">Table of n, a(n) for n = 0..1000</a>

%H Adrian Rossiter, <a href="http://www.antiprism.com/programs/waterman.html">Antiprism</a>

%H Steve Waterman, <a href="http://watermanpolyhedron.com/polyhedra.html">Polyhedra Project</a>

%H Steve Waterman, <a href="http://watermanpolyhedron.com/MENSrhdZZZ.html">Waterman's Polyhedral Mensuration chart</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

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

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

%F a(n) = 16*n+a(n-1), a(0)=4. - _Vincenzo Librandi_, Nov 13 2010

%F a(n) = A069129(n+1) + 3. - _Omar E. Pol_, Sep 04 2011

%F a(n) = A035008(n) + 4. - _Omar E. Pol_, Jun 12 2014

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

%t 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 *)

%o (PARI) a(n)=8*n^2+8*n+4 \\ _Charles R Greathouse IV_, Jul 17 2011

%o (MAGMA) [ 8*n^2 + 8*n + 4 : n in [0..50] ]; // _Wesley Ivan Hurt_, Jun 09 2014

%Y Cf. A016742.

%K nonn,easy

%O 0,1

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

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Last modified November 21 14:18 EST 2019. Contains 329371 sequences. (Running on oeis4.)