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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.

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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.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 22, 32, 46, 65, 90, 123, 167, 226, 305, 410, 549, 733, 977, 1301, 1731, 2301, 3056, 4056, 5381, 7137, 9464, 12547, 16631, 22041, 29208, 38703, 51282, 67946, 90021, 119264, 158003, 209322, 277306, 367366, 486670, 644714, 854078, 1131427
Offset: 0

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Author

Enrique Navarrete, May 01 2020

Keywords

Comments

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.
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).
First difference is A099559.

Examples

			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}.

Crossrefs

Cf. A099559, A145131.

Formula

Conjectures from Colin Barker, May 17 2020: (Start)
G.f.: x / ((1 - x)^2*(1 - x + x^2)*(1 - x^2 - x^3)).
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.
(End)
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