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%I #7 May 13 2016 05:30:06
%S 1,1,3,3,3,6,6,6,6,10,10,10,10,10,15,15,15,15,15,10,21,21,21,21,21,15,
%T 21,28,28,28,28,28,21,28,28,36,36,36,36,36,28,36,36,28,45,45,45,45,45,
%U 28,45,45,28,45,55,55,55,55,55,45,55,55,45,55,55,66,66,66,66,66,55,66
%N The largest part in the set of all {x,y,z} for any partitioning n=x+y+z into three triangular numbers x, y and z.
%C Write down all possible n=A000217(i)+A000217(j)+A000217(k) with 0<=i<=j<=k and set a(n) to the maximum term found on the right hand sides.
%H Giovanni Resta, <a href="/A171601/b171601.txt">Table of n, a(n) for n = 1..10000</a>
%e n=8: 8=1+1+6 =A000217(1)+A000217(1)+A000217(3), admits only one partitioning, so a(8)=max(1,6)=6.
%e n=17: 17 = 1+1+15 = 1+6+10. a(17) = max(1,15,6,10) = 15.
%e n=18: 18= 0+3+15 = 6+6+6. a(18) = max(0,3,15,6) = 15.
%e n=20: 20= 0+10+10 (one partitioning only), a(20)= max(0,10)=10.
%e n=21: 21= 1+10+10 = 0+0+21 = 0+6+15 = 3+3+15. a(21) = max(1,10,0,21,6,15,3) = 21.
%t a[n_] := Max@ Flatten@ IntegerPartitions[ n, {3}, (# (# + 1)/2) & /@ Range[0, Sqrt[8*n + 1]/2]]; Array[a, 100] (* _Giovanni Resta_, May 12 2016 *)
%K easy,nonn
%O 1,3
%A _Tanya Khovanova_ and _R. J. Mathar_, Dec 12 2009
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