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Every nonnegative integer can be written (eventually in many ways) as a sum of three triangular numbers by the Gauss Eureka theorem.

What is the smallest positive integer $n=n_m$ which can not be written in the form $$n=\binom{a}{2}+\binom{b}{2}+\binom{c}{2}.$$ subject to $\max(a,b,c) \le m?$

The answers for $m$ from $1$ to 60 are

$\begin {array}{cccccccccc} 1&4&8&11&24&29&29&47&68&68 \\95&99&137&141&173&173&245&281&314&314 \\314&407&419&419&470&470&617&617&711&800 \\863&911&911&911&911&1118&1118&1118&1118&1118 \\1383&1433&1433&1679&1679&1679&1868&1868&1868&1868 \\1868&2360&2493&2493&2519&2925&3044&3044&3098&3098 \end {array}$

The sequence does not seem to be in the OEIS even with a superseeker search. It just seems a curious sequence. Any information would be welcome. What bounds or asymptotics can be established? It would appear that $n_m \lt m^2$ although $m_{105}=11018 \lt 11025=105^2$ leaves some doubt. I'd conjecture (rashly) that $$\limsup \frac{n_m}{m^2}=1$$ but $n_{110}=n_{111}=n_{112}=n_{113}=11625$ and $\frac{11625}{113^3} \approx 0.91$ so the $\liminf$ might be less. I wonder what explains the repeated values and what can be said about them.

UPDATE Noam makes a nice argument that $\lim \frac{n_m}{m^2}=3/2.$ If I'd accept it if it was an answer and not a comment. Let me spell out that $8\binom{t}{2}+1=(2t+1)^2$ so $n=\binom{a}{2}+\binom{b}{2}+\binom{c}{2}$ exactly if $8n+3=(2a+1)^2+(2b+1)^2+(2c+1)^2$ and this allows one to pull in results on sums of squares.

I still don't see why things such as $n_{36}=n_{37}=n_{38}=n_{39}=n_{40}$ happen, then a revised question could be but I have not thought about it very deeply.

For any fixed $\lim k$, $n_m \frac{n_m-3m^2/2}{m}.$ The largest lt \frac{3m^2}{2}-km$with finitely many exceptions. so one could wonder about things which can be expressed are$3n^2/2-3n/2,3/2n^2-5n/2+1$and like$3n^2/2-7n/2+3.$But \frac{3m^2}{2}-\sqrt{m^3}.$ However this result was great for the problem I gather wanted to apply it to. I would have mentioned the connection sooner but the answer came before I ggot to thateventually we can get everything up past $1.49n^2.$.

3 added 312 characters in body

Every nonnegative integer can be written (eventually in many ways) as a sum of three triangular numbers by the Gauss Eureka theorem.

What is the smallest positive integer $n=n_m$ which can not be written in the form $$n=\binom{a}{2}+\binom{b}{2}+\binom{c}{2}.$$ subject to $\max(a,b,c) \le m?$

The answers for $m$ from $1$ to 60 are

$\begin {array}{cccccccccc} 1&4&8&11&24&29&29&47&68&68 \\95&99&137&141&173&173&245&281&314&314 \\314&407&419&419&470&470&617&617&711&800 \\863&911&911&911&911&1118&1118&1118&1118&1118 \\1383&1433&1433&1679&1679&1679&1868&1868&1868&1868 \\1868&2360&2493&2493&2519&2925&3044&3044&3098&3098 \end {array}$

The sequence does not seem to be in the OEIS even with a superseeker search. It just seems a curious sequence. Any information would be welcome. What bounds or asymptotics can be established? It would appear that $n_m \lt m^2$ although $m_{105}=11018 \lt 11025=105^2$ leaves some doubt. I'd conjecture (rashly) that $$\limsup \frac{n_m}{m^2}=1$$ but $n_{110}=n_{111}=n_{112}=n_{113}=11625$ and $\frac{11625}{113^3} \approx 0.91$ so the $\liminf$ might be less. I wonder what explains the repeated values and what can be said about them.

UPDATE Noam makes a nice argument that $\lim \frac{n_m}{m^2}=3/2.$ If so, then a revised question could be about $\lim \frac{n_m-3m^2/2}{m}.$ The largest things which can be expressed are $3n^2/2-3n/2,3/2n^2-5n/2+1$ and $3n^2/2-7n/2+3.$ But I gather that eventually we can get everything up past $1.49n^2.$

What is the smallest positive integer $n=n_m$ which can not be written in the form $$n=\binom{a}{3}+\binom{b}{3}+\binom{c}{3}.$$ $n=\binom{a}{2}+\binom{b}{2}+\binom{c}{2}.$$subject to \max(a,b,c) \le m? The answers for m from 1 to 60 are \begin {array}{cccccccccc} 1&4&8&11&24&29&29&47&68&68 \\95&99&137&141&173&173&245&281&314&314 \\314&407&419&419&470&470&617&617&711&800 \\863&911&911&911&911&1118&1118&1118&1118&1118 \\1383&1433&1433&1679&1679&1679&1868&1868&1868&1868 \\1868&2360&2493&2493&2519&2925&3044&3044&3098&3098 \end {array} The sequence does not seem to be in the OEIS even with a superseeker search. It just seems a curious sequence. Any information would be welcome. What bounds or asymptotics can be established? It would appear that n_m \lt m^2 although m_{105}=11018 \lt 11025=105^2 leaves some doubt. I'd conjecture (rashly) that$$\limsup \frac{n_m}{m^2}=1$$but$n_{110}=n_{111}=n_{112}=n_{113}=11625$and$\frac{11625}{113^3} \approx 0.91$so the$\liminf\$ might be less. I wonder what explains the repeated values and what can be said about them.