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 A127873 #40 Jul 12 2025 22:41:51
%S A127873 8,19,39,71,118,183,269,379,516,683,883,1119,1394,1711,2073,2483,2944,
%T A127873 3459,4031,4663,5358,6119,6949,7851,8828,9883,11019,12239,13546,14943,
%U A127873 16433,18019,19704,21491,23383,25383,27494,29719,32061,34523,37108
%N A127873 a(n) = (n^3)/2 + (3*n^2)/2 + 3*n + 3.
%C A127873 Generating polynomial is Schur's polynomial of degree 3. Schur's n-degree polynomials are the first n terms of the series expansion of the e^x function. All polynomials are irreducible and belong to the An alternating Galois transitive group if n is divisible by 4 or to the Sn symmetric Galois Group otherwise (proof: Schur, 1930).
%C A127873 Number of terms < 10^k: 0, 1, 4, 11, 26, 57, 124, 270, 583, 1258, 2713, 5847, 12598, 27143, 58479, ... - _Muniru A Asiru_, Jan 13 2018
%C A127873 For n > 1, a(n-1) is the number of ternary strings of length n that contain at most one 1 and at most two 2s. For example, for n=3, a(2)=19 since from the 27 ternary strings of length 3 we exclude 110 (3 of this type), 112 (3 of this type), 111 and 222. - _Enrique Navarrete_, Apr 16 2025
%H A127873 Vincenzo Librandi, <a href="/A127873/b127873.txt">Table of n, a(n) for n = 1..5000</a>
%H A127873 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A127873 G.f.: x*(8-13*x+11*x^2-3*x^3)/(1-x)^4. - _Colin Barker_, Apr 17 2012
%F A127873 E.g.f.: (1+x+x^2/2)*(1+x)*exp(x) - 3x - 1. - _Enrique Navarrete_, Apr 16 2025
%p A127873 A127873 := [seq((n^3)/2+(3*n^2)/2+3*n+3,n=1..10^3)]; # _Muniru A Asiru_, Jan 13 2018
%t A127873 Table[3 + 3 x + (3 x^2)/2 + x^3/2, {x, 41}]
%t A127873 Rest@ CoefficientList[ Series[-x (3 x^3 -11 x^2 +13 x - 8)/(x -1)^4, {x, 0, 41}], x] (* or *)
%t A127873 LinearRecurrence[{4, -6, 4, -1}, {8, 19, 39, 71}, 41] (* _Robert G. Wilson v_, Jan 06 2018 *)
%o A127873 (PARI) a(n)=n^3/2+3*n*(n+2)/2+3 \\ _Charles R Greathouse IV_, May 15 2013
%o A127873 (GAP) A127873 := List([1..10^3],n->(n^3)/2+(3*n^2)/2+3*n+3); # _Muniru A Asiru_, Jan 13 2018
%o A127873 (Magma) [(n^3)/2 + (3*n^2)/2 + 3*n + 3: n in [0..50]]; // _Vincenzo Librandi_, Jan 14 2018
%o A127873 (Python)
%o A127873 def A127873(n): return (n*(n*(n+3)+6)>>1)+3 # _Chai Wah Wu_, Jul 12 2025
%Y A127873 Cf. A127874.
%K A127873 nonn,easy
%O A127873 1,1
%A A127873 _Artur Jasinski_, Feb 04 2007