A226084 Number of partitions of n with Cookie Monster number 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

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

Andrew Howroyd, Table of n, a(n) for n = 1..1000

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

Cf. A000041, A226107.

Sequence in context: A001318 A024702 A343944 * A294861 A161664 A080547

Adjacent sequences: A226081 A226082 A226083 * A226085 A226086 A226087

Keyword

nonn

Author

Leigh Marie Braswell and Tanya Khovanova, May 25 2013

Status

approved