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A226084 Number of partitions of n with Cookie Monster number 2. 2
0, 0, 1, 2, 5, 7, 12, 16, 21, 29, 35, 43, 51, 66, 68, 88, 92, 117, 117, 145, 146, 185, 176, 223, 207, 267, 254, 310, 287, 368, 330, 416, 392, 476, 418, 555, 477, 603, 560, 669, 590, 770, 651, 829, 753, 902, 782, 1039, 846, 1071 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
Given a set of integers representing the number of cookies in jars, The Cookie Monster number is the minimum number of moves Cookie Monster must use to empty the jars when in one move he may choose any subset of jars and take the same number of cookies from each of those jars.
Partitions have Cookie Monster number 2 if either they have two distinct values, or they have three distinct values, where the largest value is the sum of the other two.
LINKS
L. M. Braswell and T. Khovanova, Cookie Monster Devours Naccis, arXiv:1305.4305 [math.HO], 2013.
EXAMPLE
If there are 7 cookies, the total number of partitions is 15. Two partitions, (1,1,1,1,1,1,1) and (7), correspond to Cookie Monster number 1 (they have one value). One partition (1,2,4) has Cookie Monster number 3 (it has three values and the largest is not the sum of the other two). Other partitions have Cookie Monster number 2, so a(7)=12.
MATHEMATICA
Table[Length[
Select[IntegerPartitions[n],
Length[Union[#]] ==
2 || (Length[Union[#]] == 3 &&
Union[#][[3]] == Union[#][[1]] + Union[#][[2]]) &]], {n, 50}]
PROG
(PARI) a(n)={sum(k=1, n-1, sumdiv(k, i, sumdiv(n-k, j, if(i<j, min(k/i, (n-k)/j)))))} \\ Andrew Howroyd, Apr 29 2020
CROSSREFS
Sequence in context: A001318 A024702 A343944 * A294861 A161664 A080547
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified May 23 05:11 EDT 2024. Contains 372758 sequences. (Running on oeis4.)