

A277045


Irregular triangle T(n,k) read by rows giving the number of partitions of length k such that all of the members of the partition are distinct and in A003586.


0



1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 2, 2, 0, 3, 1, 1, 0, 2, 3, 1, 2, 3, 1, 0, 2, 4, 1, 0, 2, 3, 2, 0, 2, 4, 3, 1, 1, 4, 2, 1, 0, 2, 4, 3, 1, 2, 4, 4, 1, 0, 2, 5, 4, 1, 0, 3, 3, 5, 1, 0, 2, 6, 5, 2, 0, 2, 5, 5, 3, 0, 0, 7, 5, 3, 1, 2, 4, 7, 3, 1, 0, 2, 5, 8, 2, 1, 0, 2, 5, 6, 5, 1
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OFFSET

1,8


COMMENTS

If n is in A003586, then T(n,1) = 1, else T(n,1) = 0.
T(n,k) also is the number of ways of representing n involving k 1's in the base(2,3) or "dualbase number system" (i.e., base(2,3)).
The number of "canonic" representations of n in a dualbase number system as defined by the reference as having the lowest number of terms, appears in the first column of the triangle with a value greater than 0.
A237442(n) = the least k with a nonzero value.


REFERENCES

V. Dimitrov, G. Jullien, R. Muscedere, Multiple Number Base System Theory and Applications, 2nd ed., CRC Press, 2012, pp. 3539.


LINKS

Table of n, a(n) for n=1..100.


EXAMPLE

Triangle starts:
1
1
1,1
1,1
0,2
1,1,1
0,2,1
1,1,1
1,2,2
0,3,1,1
0,2,3
1,2,3,1
0,2,4,1
0,2,3,2
0,2,4,3
1,1,4,2,1
0,2,4,3
1,2,4,4,1
0,2,5,4,1
0,3,3,5,1
...
Row n = 10 has terms {0,3,1,1} because 10 is not in A003586 thus k = 1 has value 0. The partitions of 10 that have distinct members that are in A003586 are {{1,9},{2,8},{4,6},{1,3,6},{1,2,3,4}}, thus there are 3 partitions of length k = 2, 1 of length k = 3, and 1 with k = 4. A237442(10) = 2.


MATHEMATICA

nn = 6^6; t = Sort@ Select[Flatten@ KroneckerProduct[2^Range[0, Ceiling@ Log2@ nn], 3^Range[0, Ceiling@ Log[3, nn]]], # <= nn &]; Table[BinCounts[#, {1, Max@ # + 1, 1}] &@ Map[Length, #] &@ Select[Subsets@ TakeWhile[t, # <= n &], Total@ # == n &], {n, 40}]


CROSSREFS

Cf. A003586, A237442, A276380.
Sequence in context: A287364 A340676 A117162 * A146061 A331186 A135936
Adjacent sequences: A277042 A277043 A277044 * A277046 A277047 A277048


KEYWORD

nonn,tabf


AUTHOR

Michael De Vlieger, Sep 27 2016


STATUS

approved



