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An easy way, using PowersRepresentations[]:

$PowersRepresentations[n,k,p] $ gives the distinct representations of the integer n as a sum of k non-negative p^th integer powers.

s1 = Join@@ (Select[PowersRepresentations[748,#,2], # ⋂ Range@16 == # &] & /@ Range@8)

Gives all possible ways to total 748.

For filtering the ones that are complementary:

s = Tally[s1, Union[#1, #2] == Range@16 &][[All, 1]];
Length@s
(*
 57
*)

An easy way, using PowersRepresentations[]:

$PowersRepresentations[n,k,p] $ gives the distinct representations of the integer n as a sum of k non-negative p^th integer powers.

s1 = Join@@ (Select[PowersRepresentations[748,#,2], # ⋂ Range@16 == # &] & /@ Range@8)

Gives all possible ways to total 748.

For filtering the ones that are complementaries:

s = Tally[s1, Union[#1, #2] == Range@16 &][[All, 1]];
Length@s
(*
 57
*)

An easy way, using PowersRepresentations[]:

$PowersRepresentations[n,k,p] $ gives the distinct representations of the integer n as a sum of k non-negative p^th integer powers.

s1 = Join@@ (Select[PowersRepresentations[748,#,2], Intersection[#, Range@16] == # &] & /@ Range@8)

Gives all possible ways to total 748.

For filtering the ones that are complementary:

s = Tally[s1, Union[#1, #2] == Range@16 &][[All, 1]];
Length@s
(*
 57
*)

An easy way, using PowersRepresentations[]:

$PowersRepresentations[n,k,p] $ gives the distinct representations of the integer n as a sum of k non-negative p^th integer powers.

s1 = Join@@ (Select[PowersRepresentations[748,#,2], # ⋂ Range@16 == # &] & /@ Range@8)

Gives all possible ways to total 748.

For filtering the ones that are complementary:

s = Tally[s1, Union[#1, #2] == Range@16 &][[All, 1]];
Length@s
(*
 57
*)

An easy way, using PowersRepresentations[]:

$PowersRepresentations[n,k,p] $ gives the distinct representations of the integer n as a sum of k non-negative p^th integer powers.

s1 = Join@@ (Select[PowersRepresentations[748,#,2], Intersection[#, Range@16] == # &] & /@ Range@8)

Gives all possible ways to total 748.

For filtering the complementary ones:

s = Tally[s1, Union[#1, #2] == Range@16 &][[All, 1]];
Length@s
(*
 57
*)

An easy way, using PowersRepresentations[]:

$PowersRepresentations[n,k,p] $ gives the distinct representations of the integer n as a sum of k non-negative p^th integer powers.

s1 = Join@@ (Select[PowersRepresentations[748,#,2], Intersection[#, Range@16] == # &] & /@ Range@8)

Gives all possible ways to total 748.

For filtering the ones that are complementary:

s = Tally[s1, Union[#1, #2] == Range@16 &][[All, 1]];
Length@s
(*
 57
*)