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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 (list; graph; refs; listen; history; text; internal format)
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 "dual-base number system" (i.e., base(2,3)).

The number of "canonic" representations of n in a dual-base 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. 35-39.

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: A090340 A287364 A117162 * A146061 A135936 A109707

Adjacent sequences:  A277042 A277043 A277044 * A277046 A277047 A277048

KEYWORD

nonn,tabf

AUTHOR

Michael De Vlieger, Sep 27 2016

STATUS

approved

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Last modified October 15 05:43 EDT 2019. Contains 328026 sequences. (Running on oeis4.)