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 A218152 #47 May 04 2023 18:07:27
%S A218152 1,2,5,13,29,56,97,155,233,334,461,617,805,1028,1289,1591,1937,2330,
%T A218152 2773,3269,3821,4432,5105,5843,6649,7526,8477,9505,10613,11804,13081,
%U A218152 14447,15905,17458,19109,20861,22717,24680,26753,28939,31241,33662,36205,38873
%N A218152 a(n) = 1 + n + ((n-1)*n^2)/2.
%C A218152 a(n) = sum(i=0,1,2,...k) d(i)*C(n,i), d(0)=a(0), C(n,i)=0 for all i > n. I would introduce the arithmetic-arithmetic sequence which is defined as the sequence of finite differences, that is, with k consecutive rows of differences, whose first terms are d(1), d(2), d(3),..., d(k), the last row (k-th row) being of a constant difference. Here, it is submitted a special case of the above mentioned sequence with k=3, d(0)=d(1)=1, d(2)=2, d(3)=3.
%C A218152 This sequence is not in Comtet. - _T. D. Noe_, Nov 16 2012
%C A218152 a(n) appears to be the number of configurations of n equilateral triangles that are allowed to have common vertices, where A002061(n) gives the number of connected configurations and A060354(n) is the number of configurations consisting of several pieces. - _Anton Zakharov_, May 13 2018
%D A218152 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72.
%H A218152 Colin Barker, <a href="/A218152/b218152.txt">Table of n, a(n) for n = 0..1000</a>
%H A218152 Anton Zakharov, <a href="/A218152/a218152.png">Illustration of initial terms</a>
%H A218152 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A218152 a(n) = a(n-1)+(4-5*n+3*n^2)/2 for n > 0 and a(0)=1.
%F A218152 a(n) = A006000(n-1)+1 for n > 0. - _Antti Karttunen_, Oct 24 2012
%F A218152 a(n) = A060354(n) + A002061(n). - _Anton Zakharov_, May 13 2018
%F A218152 G.f.: (x^3+3*x^2-2*x+1)/(x-1)^4. - _Alois P. Heinz_, May 13 2018
%F A218152 From _Colin Barker_, May 13 2018: (Start)
%F A218152 a(n) = (2 + 2*n - n^2 + n^3) / 2.
%F A218152 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3.
%F A218152 (End)
%e A218152 for n=5, a(5) = 1+5+(4*25)/2 = 1+5+100/2 = 1+5+50 = 56.
%t A218152 Table[1+n+((n-1)n^2)/2,{n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,2,5,13},50] (* _Harvey P. Dale_, May 04 2023 *)
%o A218152 (PARI) Vec((1 - 2*x + 3*x^2 + x^3) / (1 - x)^4 + O(x^40)) \\ _Colin Barker_, May 13 2018
%Y A218152 Cf. A000027, A000124, A000125, A060354, A002061.
%K A218152 nonn,easy
%O A218152 0,2
%A A218152 _Mokhtar Mohamed_, Oct 24 2012
%E A218152 Corrected and edited by _Mokhtar Mohamed_, Nov 17 2012
%E A218152 Missing term 1937 inserted by _Alois P. Heinz_, Jun 11 2017