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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. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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.
LINKS
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)
EXAMPLE
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
Sequence in context: A122047 A177100 A265071 * A226239 A209231 A137358
KEYWORD
nonn
AUTHOR
Enrique Navarrete, May 01 2020
STATUS
approved

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Last modified April 18 16:22 EDT 2024. Contains 371780 sequences. (Running on oeis4.)