#Mathematica, 53 bytes

Min[Plus@@@Table[$($+1)/2,{,ドル#+1}]~FrobeniusSolve~#]&

This code is very slow. If you want to test this function, use the following version instead:

Min[Plus@@@Table[$($+1)/2,{,ドル√#+1}]~FrobeniusSolve~#]&

Try it on Wolfram Sandbox

###Explanation

Min[Plus@@@Table[$($+1)/2,{,ドル#+1}]~FrobeniusSolve~#]& (* input: n *)
 Table[$($+1)/2,{,ドル#+1}] (* Generate the first n triangle numbers *)
 ~FrobeniusSolve~# (* Generate a Frobenius equation from the *)
 (* triangle numbers and find all solutions. *)
 Plus@@@ (* Sum each solution set *)
Min (* Fetch the smallest value *)

Mathematica, 53 bytes

Min[Plus@@@Table[$($+1)/2,{,ドル#+1}]~FrobeniusSolve~#]&

This code is very slow. If you want to test this function, use the following version instead:

Min[Plus@@@Table[$($+1)/2,{,ドル√#+1}]~FrobeniusSolve~#]&

Try it on Wolfram Sandbox

Explanation

Min[Plus@@@Table[$($+1)/2,{,ドル#+1}]~FrobeniusSolve~#]& (* input: n *)
 Table[$($+1)/2,{,ドル#+1}] (* Generate the first n triangle numbers *)
 ~FrobeniusSolve~# (* Generate a Frobenius equation from the *)
 (* triangle numbers and find all solutions. *)
 Plus@@@ (* Sum each solution set *)
Min (* Fetch the smallest value *)

#Mathematica, 53 bytes

Min[Plus@@@Table[$($+1)/2,{,ドル#+1}]~FrobeniusSolve~#]&

#Mathematica, 53 bytes

Min[Plus@@@Table[$($+1)/2,{,ドル#+1}]~FrobeniusSolve~#]&

This code is very slow. If you want to test this function, use the following version instead:

Min[Plus@@@Table[$($+1)/2,{,ドル√#+1}]~FrobeniusSolve~#]&

Try it on Wolfram Sandbox

###Explanation

Min[Plus@@@Table[$($+1)/2,{,ドル#+1}]~FrobeniusSolve~#]& (* input: n *)
 Table[$($+1)/2,{,ドル#+1}] (* Generate the first n triangle numbers *)
 ~FrobeniusSolve~# (* Generate a Frobenius equation from the *)
 (* triangle numbers and find all solutions. *)
 Plus@@@ (* Sum each solution set *)
Min (* Fetch the smallest value *)