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 A133263 #51 Jun 02 2025 04:55:47
%S A133263 1,3,5,8,12,17,23,30,38,47,57,68,80,93,107,122,138,155,173,192,212,
%T A133263 233,255,278,302,327,353,380,408,437,467,498,530,563,597,632,668,705,
%U A133263 743,782,822,863,905,948,992,1037,1083,1130,1178,1227,1277,1328,1380,1433
%N A133263 Binomial transform of (1, 2, 0, 1, -1, 1, -1, 1, ...).
%C A133263 A007318 * [1, 2, 0, 1, -1, 1, -1, 1, ...]. Left column of A134249.
%H A133263 Harvey P. Dale, <a href="/A133263/b133263.txt">Table of n, a(n) for n = 0..1000</a>
%H A133263 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A133263 From _Emeric Deutsch_, Nov 12 2007: (Start)
%F A133263 a(n) = (n^2 + n + 4)/2 for n > 0.
%F A133263 G.f.: (1 - x^2 + x^3)/(1-x)^3. (End)
%F A133263 a(n) = A000124(n) + 1, n >= 1. - _Zerinvary Lajos_, Apr 12 2008
%F A133263 a(0)=1, a(1)=3; for n >= 2, a(n) = a(n-1) + n. - Philippe Lallouet (philip.lallouet(AT)orange.fr), May 27 2008; corrected by _Michel Marcus_, Nov 03 2018
%F A133263 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=3, a(2)=5, a(3)=8. - _Harvey P. Dale_, Feb 13 2012
%F A133263 a(n) = A238531(n+1) if n >= 0. - _Michael Somos_, Feb 28 2014
%F A133263 For n > 0: A228446(a(n)) = 5. - _Reinhard Zumkeller_, Mar 12 2014
%F A133263 a(n) = A022856(n+4) for n >= 1. - _Georg Fischer_, Nov 02 2018
%F A133263 Sum_{n>=0} 1/a(n) = 1/2 + 2*Pi*tanh(sqrt(15)*Pi/2)/sqrt(25). - _Amiram Eldar_, Jun 02 2025
%e A133263 a(3) = 8 = (1, 3, 3, 1) dot (1, 2 0, 1) = (1 + 6 + 0 + 1).
%p A133263 1, seq((n^2+n+4)*1/2,n=1..50); # _Emeric Deutsch_, Nov 12 2007
%p A133263 a:=n->add((Stirling2(j+1,n)), j=0..n): seq(a(n)+1, n=0..50); # _Zerinvary Lajos_, Apr 12 2008
%t A133263 Join[{1},Table[(n^2+n+4)/2,{n,50}]] (* or *) Join[{1}, LinearRecurrence[ {3,-3,1},{3,5,8},50]] (* _Harvey P. Dale_, Feb 13 2012 *)
%o A133263 (PARI) a(n)=n*(n+1)/2+2 \\ _Charles R Greathouse IV_, Mar 26 2014
%Y A133263 Cf. A000124, A007318, A022856, A134249, A228446, A238531.
%K A133263 nonn,easy
%O A133263 0,2
%A A133263 _Gary W. Adamson_, Oct 15 2007
%E A133263 More terms from _Emeric Deutsch_, Nov 12 2007