This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A004188 #56 Sep 08 2022 08:44:32
%S A004188 0,1,11,39,94,185,321,511,764,1089,1495,1991,2586,3289,4109,5055,6136,
%T A004188 7361,8739,10279,11990,13881,15961,18239,20724,23425,26351,29511,
%U A004188 32914,36569,40485,44671,49136,53889,58939,64295,69966,75961
%N A004188 a(n) = n*(3*n^2 - 1)/2.
%C A004188 3-dimensional analog of centered polygonal numbers.
%C A004188 (1), (4+7), (10+13+16), (19+22+25+28), ... - _Jon Perry_, Sep 10 2004
%D A004188 E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.
%D A004188 T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
%H A004188 Vincenzo Librandi, <a href="/A004188/b004188.txt">Table of n, a(n) for n = 0..5000</a>
%H A004188 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A004188 Partial sums of n-1 3-spaced triangular numbers, e.g., a(4) = t(1) + t(4) + t(7) = 1 + 10 + 28 = 39. - _Jon Perry_, Jul 23 2003
%F A004188 a(n) = C(2*n+1,3) + C(n+1,3), n >= 0. - _Zerinvary Lajos_, Jan 21 2007
%F A004188 a(n) = A000447(n) + A000292(n). - _Zerinvary Lajos_, Jan 21 2007
%F A004188 G.f.: x*(1+7*x+x^2) / (x-1)^4. - _R. J. Mathar_, Oct 08 2011
%F A004188 From _Miquel Cerda_, Dec 25 2016: (Start)
%F A004188 a(n) = A000578(n) + A135503(n).
%F A004188 a(n) = A007588(n) - A135503(n). (End)
%F A004188 E.g.f.: (x/2)*(2 + 9*x + 3*x^2)*exp(x). - _G. C. Greubel_, Sep 01 2017
%p A004188 seq(binomial(2*n+1,3)+binomial(n+1,3), n=0..37); # _Zerinvary Lajos_, Jan 21 2007
%t A004188 Table[n(3n^2-1)/2,{n,0,80}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 18 2011 *)
%t A004188 LinearRecurrence[{4,-6,4,-1},{0,1,11,39},40] (* _Harvey P. Dale_, Jul 19 2019 *)
%o A004188 (PARI) vector(40, n, n*(3*n^2-1)/2)
%o A004188 (Magma) [n*(3*n^2-1)/2: n in [0..50]]; //_Vincenzo Librandi_, May 15 2011
%Y A004188 1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
%Y A004188 Cf. A002412, A016061, A051682.
%Y A004188 Cf. A236770 (partial sums).
%K A004188 nonn,easy
%O A004188 0,3
%A A004188 Albert D. Rich (Albert_Rich(AT)msn.com)