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a(n-5) is the number of nonempty subsets of {1,2,...,n} such that the difference of successive elements is at least 5.
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%I #14 Jun 24 2020 04:32:54

%S 0,1,3,6,10,15,22,32,46,65,90,123,167,226,305,410,549,733,977,1301,

%T 1731,2301,3056,4056,5381,7137,9464,12547,16631,22041,29208,38703,

%U 51282,67946,90021,119264,158003,209322,277306,367366,486670,644714,854078,1131427

%N 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 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 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 First difference is A099559.

%F Conjectures from _Colin Barker_, May 17 2020: (Start)

%F G.f.: x / ((1 - x)^2*(1 - x + x^2)*(1 - x^2 - x^3)).

%F 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 (End)

%e 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 Cf. A099559, A145131.

%K nonn

%O 0,3

%A _Enrique Navarrete_, May 01 2020