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 A134869 #62 Jun 02 2025 04:58:08
%S A134869 1,4,7,11,16,22,29,37,46,56,67,79,92,106,121,137,154,172,191,211,232,
%T A134869 254,277,301,326,352,379,407,436,466,497,529,562,596,631,667,704,742,
%U A134869 781,821,862,904,947,991,1036,1082,1129,1177,1226,1276,1327,1379,1432
%N A134869 Row sums of triangle A134868.
%C A134869 Where records occur in A182703. - _Omar E. Pol_, Feb 14 2012
%C A134869 Consider quadratic polynomials x^2+cx+d. Then a(n) is the number of these polynomials with 0 <= c < n, 0 <= d < n where no polynomial can be horizontally translated into another. For example, a(3) = 7, the coefficients are as follows: (c, d) = {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0)}. Two polynomials are excluded, namely x^2+2x+1 = (x+1)^2+0(x+1)+0, and x^2+2x+2 = (x+1)^2+0(x+1)+1. - _Griffin N. Macris_, Jul 19 2016
%C A134869 a(n) gives the number of regions into which the square [0,1]x[0,1] is divided by the Bernstein polynomials of degree n. - _Franck Maminirina Ramaharo_, Feb 28 2018
%H A134869 Michael De Vlieger, <a href="/A134869/b134869.txt">Table of n, a(n) for n = 1..10000</a>
%H A134869 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A134869 a(n) = 1, then for n>1, a(n) = T(n) + 1, where A000217 = (1, 3, 6, 10, 15, ...).
%F A134869 Binomial transform of [1, 3, 0, 1, -1, 1, -1, 1, ...].
%F A134869 From _R. J. Mathar_, Oct 27 2008: (Start)
%F A134869 G.f.: x(1+x-2x^2+x^3)/(1-x)^3.
%F A134869 a(n) = 1 + A000217(n) = A000124(n), n > 1. (End)
%F A134869 Sum_{n>=1} 1/a(n) = 2*Pi*tanh(sqrt(7)*Pi/2)/sqrt(7) - 1/2. - _Amiram Eldar_, Jun 02 2025
%e A134869 a(4) = 11 = sum of row 4 terms of triangle A134868: (2, + 2 + 3 + 4).
%e A134869 a(4) = 11 = 1 + 10, where 10 = T(4).
%e A134869 a(4) = 11 = (1, 3, 3, 1) dot (1, 3, 0, 1) = (1 + 9 + 0 + 1).
%p A134869 a:=n->sum((stirling2(j+1,n)), j=1..n):seq(a(n), n=1..50); # _Zerinvary Lajos_, Apr 12 2008
%t A134869 Table[(n^2 + n)/2 + Boole[n != 1], {n, 53}] (* or *)
%t A134869 Table[PolygonalNumber@ n + Boole[n != 1], {n, 53}] (* Version 10.4, or *)
%t A134869 Table[Sum[StirlingS2[k + 1, n], {k, n}], {n, 53}] (* or *)
%t A134869 Rest@ CoefficientList[Series[x (1 + x - 2 x^2 + x^3)/(1 - x)^3, {x, 0, 53}], x] (* _Michael De Vlieger_, Jul 19 2016 *)
%o A134869 (PARI) a(n)=if(n>1, n*(n+1)/2+1, 1) \\ _Charles R Greathouse IV_, Aug 05 2016
%Y A134869 Cf. A000217, A134868, A182703.
%Y A134869 Essentially the same as A000124.
%K A134869 nonn,easy
%O A134869 1,2
%A A134869 _Gary W. Adamson_, Nov 14 2007