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a(n) is the smallest positive integer which can be represented as the sum of distinct positive triangular numbers in exactly n ways, or 0 if no such integer exists.
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%I #22 Jun 09 2016 04:42:39

%S 1,10,25,31,49,46,55,67,70,76,82,117,102,91,97,107,101,135,110,112,

%T 116,115,119,128,0,131,133,130,148,145,136,0,137,149,154,146,0,169,

%U 152,157,155,168,171,158,174,161,0,183,184,167,0,0,173,0,175,181,190

%N a(n) is the smallest positive integer which can be represented as the sum of distinct positive triangular numbers in exactly n ways, or 0 if no such integer exists.

%C 46 is the smallest number that can be expressed as the sum of distinct triangular numbers in five ways, but 49 is the smallest that can be so expressed in _exactly_ five ways. There are further examples of this phenomenon.

%e 25 = 1 + 3 + 6 + 15 = 10 + 15 = 1 + 3 + 21. This is the smallest number that can be written as the sum of distinct triangular numbers in three different ways. So a(3)=25.

%e The first null values of a(n) occur for n = 25, 32, 37, 47, 51, 52, 54, 61,... - _Giovanni Resta_, Jun 08 2016

%t nT[n_, m_: 0] := nT[n,m] = If[n == 0, 1, Block[{t, i=m+1, s=0}, While[(t = i*(i+1)/2) <= n, s += nT[n-t, i]; i++]; s]]; a[n_] := Block[{k=0, t}, While[(t = nT[++k]) != n && t < Max[2*n, 30]]; If[t == n, k, 0]]; Array[a, 57] (* _Giovanni Resta_, Jun 08 2016 *)

%Y Cf. A007294, A060773, A024940, A064816.

%K nonn

%O 1,2

%A _Phil Scovis_, Jun 07 2016

%E a(15)-a(20) from _Tom Edgar_, Jun 08 2016

%E a(21)-a(57) from _Giovanni Resta_, Jun 08 2016