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%I #8 Jan 10 2018 20:28:33
%S 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,
%T 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,
%U 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
%N 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.
%C If n is in A003586, then T(n,1) = 1, else T(n,1) = 0.
%C 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)).
%C 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.
%C A237442(n) = the least k with a nonzero value.
%D V. Dimitrov, G. Jullien, R. Muscedere, Multiple Number Base System Theory and Applications, 2nd ed., CRC Press, 2012, pp. 35-39.
%e Triangle starts:
%e 1
%e 1
%e 1,1
%e 1,1
%e 0,2
%e 1,1,1
%e 0,2,1
%e 1,1,1
%e 1,2,2
%e 0,3,1,1
%e 0,2,3
%e 1,2,3,1
%e 0,2,4,1
%e 0,2,3,2
%e 0,2,4,3
%e 1,1,4,2,1
%e 0,2,4,3
%e 1,2,4,4,1
%e 0,2,5,4,1
%e 0,3,3,5,1
%e ...
%e 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.
%t 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}]
%Y Cf. A003586, A237442, A276380.
%K nonn,tabf
%O 1,8
%A _Michael De Vlieger_, Sep 27 2016
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