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A114091
Number of partitions of n into parts that are distinct mod 3.
1
1, 1, 2, 2, 2, 4, 3, 3, 7, 4, 4, 11, 5, 5, 16, 6, 6, 22, 7, 7, 29, 8, 8, 37, 9, 9, 46, 10, 10, 56, 11, 11, 67, 12, 12, 79, 13, 13, 92, 14, 14, 106, 15, 15, 121, 16, 16, 137, 17, 17, 154, 18, 18, 172, 19, 19, 191, 20, 20, 211, 21, 21, 232, 22, 22, 254, 23, 23, 277, 24
OFFSET
1,3
COMMENTS
Each partition can have at most three parts if n is a multiple of three and at most two parts otherwise. - Andrew Howroyd, Jan 28 2020
In general, these sequences can be generated by a linear recurrence with a signature that contains k=1..d tuples of the form (d-1 zeros, (-1)^(k-1) * binomial(d, k)), where d = number of distinct parts (here: d=3). - Georg Fischer, Sep 03 2022
FORMULA
a(3*n) = 1 + n + binomial(n, 2); a(3*n-1) = a(3*n-2) = n. - Andrew Howroyd, Jan 28 2020
EXAMPLE
a(5)=2 because there are 2 such partition of 5: {5}, {2,3}.
MATHEMATICA
<< DiscreteMath`Combinatorica`; np[n_]:= Length@Select[Mod[ #, 3]& /@ Partitions[n], (Length@# != Length@Union@#)&]; lst = Array[np, 50] (* or *)
LinearRecurrence[{0, 0, 3, 0, 0, -3, 0, 0, 1}, {1, 1, 2, 2, 2, 4, 3, 3, 7}, 64] (* Georg Fischer, Sep 03 2022 *)
PROG
(PARI) a(n)={1 + n\3 + if(n%3==0, binomial(n/3, 2))} \\ Andrew Howroyd, Jan 28 2020
CROSSREFS
Cf. A008619(d=2), A114092(4), A114093(5), A114094(6), A114095(7), A114096(8), A114098(9), A114097(10).
Sequence in context: A338796 A085454 A083403 * A209580 A166008 A194291
KEYWORD
nonn
AUTHOR
Giovanni Resta, Feb 06 2006
EXTENSIONS
Terms a(51) and beyond from Andrew Howroyd, Jan 28 2020
STATUS
approved