Pythagoras Theorem Combinatorial Number Setand Generalization

WegetPythagorasTheoremontwocombinatoiralnumbersets.If 32 +42 =52 ;62 +82 + 2 4 2 =2 62 thenwehave (ⅰ )and (ⅱ ) : (ⅰ )Ifnisanoddthen ∑[(n+ 3 ) / 2 ] t=0 n+ 3 -t t 2 -∑[(n+ 1 ) / 2 ] t=0 n+ 1 -t t 2 2 + 2 ∑[n/ 2 ] t=0 n-t t · ∑[(n+ 4 ) / 2 ] t=0 n+ 4 -t t 2 + 4 ∑[(n+ 2 ) / 2 ] t=0 n+ 2 -t t 2 =∑[(n+ 1 ) / 2 ] t=0 n +1-t t 2 +∑[(n+ 3 ) / 2 ] t=0 n +3-t t 2 2 . (ⅱ )Ifnisaneventhen ∑[(n+ 4 ) / 2 ] t=0 n+ 4 -t t 2 -∑[n/ 2 ] t=0 n-t t 2 2 + 2 ∑[(n+ 1 ) / 2 ] t=0 n+ 1 -t t · ∑[(n+ 3 ) / 2 ] t=0 n + 3 -t t 2 + 4 ∑[(n+ 2 ) / 2 ] t=0 n+ 2 -t t 2 =∑[n/ 2 ] t=0 n -t t 2 +∑[(n+ 4 ) / 2 ] t=0 n +4-t t 2 2 .