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 A330910 #14 Jun 24 2020 04:32:54
%S A330910 0,1,3,6,10,15,22,32,46,65,90,123,167,226,305,410,549,733,977,1301,
%T A330910 1731,2301,3056,4056,5381,7137,9464,12547,16631,22041,29208,38703,
%U A330910 51282,67946,90021,119264,158003,209322,277306,367366,486670,644714,854078,1131427
%N A330910 a(n-5) is the number of nonempty subsets of {1,2,...,n} such that the difference of successive elements is at least 5.
%C A330910 For n >=0 the sequence contains the triangular numbers; for n >= 5 have to add the tetrahedral numbers; for n >= 10 have to add the numbers binomial(n,4) (starting with 0,1,5,...); for n >= 15 have to add the numbers binomial(n,5) (starting with 0,1,6,..); in general, for n >= 5*k have to add to the sequence the numbers binomial(n, k+2), k >= 0.
%C A330910 For example, a(19) = 190+560+495+56, where 190 is a triangular number, 560 is a tetrahedral number, 495 is a number binomial(n,4) and 56 is a number binomial(m,5) (with the proper n, m due to shifts in the names of the sequences).
%C A330910 First difference is A099559.
%F A330910 Conjectures from _Colin Barker_, May 17 2020: (Start)
%F A330910 G.f.: x / ((1 - x)^2*(1 - x + x^2)*(1 - x^2 - x^3)).
%F A330910 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-5) - 2*a(n-6) + a(n-7) for n>6.
%F A330910 (End)
%e A330910 For example, for n=11, a(6) = 22 and the sets are: {1,6}, {1,7}, {1,8}, {1,9}, {1,10}, {1,11}, {2,7}, {2,8}, {2,9}, {2,10}, {2,11}, {3,8}, {3,9}, {3,10}, {3,11}, {4,9}, {4,10}, {4,11}, {5,10}, {5,11}, {6,11}, {1,6,11}.
%Y A330910 Cf. A099559, A145131.
%K A330910 nonn
%O A330910 0,3
%A A330910 _Enrique Navarrete_, May 01 2020