Rewrite code to make it faster

There seems to be a pattern between the size of input matrix (m) and the sum of elements in MatrixPower[m, 12] that can be discovered cheaply using FindSequenceFunction on a short list of small sizes:

(list = With[{mm = ArrayReshape[Range[#^2], {#, #}]}, 
        Total[MatrixPower[mm, 12], 2]] & /@ Range[40];) // RepeatedTiming // First

 0.11

ClearAll[seqF]
(seqF[k_] := Evaluate @ FindSequenceFunction[list, k])//RepeatedTiming // First

0.0040

seqF[k]

seqF[270] evaluates almost instantly:

(270^5)^12 seqF[270] // RepeatedTiming

Compare with

Total @ Flatten @ MatrixPower[Partition[270^5*Range[270^2], 270], 12] // 
 RepeatedTiming