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%I M2531 N0999 #50 Feb 01 2022 01:35:44
%S 0,0,1,3,6,9,15,18,27,30,45,42,66,63,84,84,120,99,153,132,174,165,231,
%T 180,270,234,297,270,378,276,435,360,450,408,540,414,630,513,636,552,
%U 780,558,861,690,828,759,1035,744,1113,870,1104,972,1326,945,1380,1116,1386,1218
%N Number of compositions of n into 3 ordered relatively prime parts.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A000741/b000741.txt">Table of n, a(n) for n = 1..10000</a>
%H H. W. Gould, <a href="http://www.fq.math.ca/Scanned/2-4/gould.pdf">Binomial coefficients, the bracket function and compositions with relatively prime summands</a>, Fib. Quart. 2(4) (1964), 241-260.
%H C. Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Kimberling/kimberling24.html">Matrix Transformations of Integer Sequences</a>, J. Integer Seqs., Vol. 6, 2003.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F Moebius transform of A000217(n-2).
%F G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = (1 - 3*x + 3*x^2)/(1 - x)^3. - _Ilya Gutkovskiy_, Apr 26 2017
%e From _Gus Wiseman_, Oct 14 2020: (Start)
%e The a(3) = 1 through a(8) = 18 triples:
%e (1,1,1) (1,1,2) (1,1,3) (1,1,4) (1,1,5) (1,1,6)
%e (1,2,1) (1,2,2) (1,2,3) (1,2,4) (1,2,5)
%e (2,1,1) (1,3,1) (1,3,2) (1,3,3) (1,3,4)
%e (2,1,2) (1,4,1) (1,4,2) (1,4,3)
%e (2,2,1) (2,1,3) (1,5,1) (1,5,2)
%e (3,1,1) (2,3,1) (2,1,4) (1,6,1)
%e (3,1,2) (2,2,3) (2,1,5)
%e (3,2,1) (2,3,2) (2,3,3)
%e (4,1,1) (2,4,1) (2,5,1)
%e (3,1,3) (3,1,4)
%e (3,2,2) (3,2,3)
%e (3,3,1) (3,3,2)
%e (4,1,2) (3,4,1)
%e (4,2,1) (4,1,3)
%e (5,1,1) (4,3,1)
%e (5,1,2)
%e (5,2,1)
%e (6,1,1)
%e (End)
%p with(numtheory):
%p mobtr:= proc(p)
%p proc(n) option remember;
%p add(mobius(n/d)*p(d), d=divisors(n))
%p end
%p end:
%p A000217:= n-> n*(n+1)/2:
%p a:= mobtr(n-> A000217(n-2)):
%p seq(a(n), n=1..58); # _Alois P. Heinz_, Feb 08 2011
%t mobtr[p_] := Module[{f}, f[n_] := f[n] = Sum[MoebiusMu[n/d]*p[d], {d, Divisors[n]}]; f]; A000217[n_] := n*(n+1)/2; a = mobtr[A000217[#-2]&]; Table[a[n], {n, 1, 58}] (* _Jean-François Alcover_, Mar 12 2014, after _Alois P. Heinz_ *)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],GCD@@#==1&]],{n,0,30}] (* _Gus Wiseman_, Oct 14 2020 *)
%Y A000010 is the length-2 version.
%Y A000217(n-2) does not require relative primality.
%Y A000740 counts these compositions of any length.
%Y A000742 is the length-4 version.
%Y A000837 counts relatively prime partitions.
%Y A023023 is the unordered version.
%Y A101271 is the strict case.
%Y A101391 has this as column k = 3.
%Y A284825*6 is the pairwise non-coprime case.
%Y A291166 intersected with A014311 ranks these compositions.
%Y A337461 is the pairwise coprime instead of relatively prime version.
%Y A337603 counts length-3 compositions whose distinct parts are pairwise coprime.
%Y A337604 is the pairwise non-coprime instead of relatively prime version.
%Y Cf. A001399, A007997, A023022, A078374, A337450, A337451, A337602.
%K nonn,easy
%O 1,4
%A _N. J. A. Sloane_
%E Edited by _Alois P. Heinz_, Feb 08 2011
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