Some additive relations in the Pascal triangle

SOME ADDITIVE RELA TIONS IN THE P ASCAL TRIANGLE A. V. STOY ANOVSKY Abstract. W e derive some, see mingly new, curious additive rela- tions in the Pascal triangle. They arise in summing up the num ber s in the triangle along some vertical line up to some place. There are a lot of kno wn additive relations among ( q -)binomial co ef- ficien ts, se e, for example, [1]. In this pap er w e deriv e a see mingly new curious relatio n obtained in a simplest wa y , namely , by summation of the n um b ers of t he P ascal tria ngle along a v ertical line up to some place. It is w ell known that the sum of the n um b ers of the P ascal t r ia ngle, 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 . . . along the n - th horizon tal line equals 2 n (see, for ex ample, [2]), (1)  n 0  +  n 1  +  n 2  + . . . +  n n  = 2 n . It is also kno wn that the sum along the k -t h diagonal up to some place equals the nex t n um b er in the next diagonal, (2)  k k  +  k + 1 k  +  k + 2 k  + . . . +  n k  =  n + 1 k + 1  . Another w ell kno wn fact is that the sum along the n -th diago nal with the slope 1 / 3 equals the n -th Fib onacci n um b er, (3)  n 0  +  n − 1 1  +  n − 2 2  +  n − 3 3  + . . . = u n , where (4) u n +2 = u n +1 + u n , u 0 = u 1 = 1 1 2 A. V. S TOY ANOVSKY are the Fib onacci n um b ers. The main purp ose o f this note is t o deriv e a form ula for the sum along a ve rtical line up to some place, (5)  n k  +  n − 2 k − 1  +  n − 4 k − 2  + . . . . This form ula is the f o llo wing. Theorem. The sum (5) equals the alternated sum along the next diagonal with the slope 1 / 3, starting f rom the closest num b er, plus p ossibly ± 1 dep ending on the vertical: (6)  n k  +  n − 2 k − 1  +  n − 4 k − 2  + . . . =  n + 1 k + 1  −  n k + 2  +  n − 1 k + 3  − . . . +        0 , n − 2 k ≤ 0 , 0 , n − 2 k > 0 , n − 2 k = 6 p, 6 p + 3 , − 1 , n − 2 k > 0 , n − 2 k = 6 p + 1 , 6 p + 2 , 1 , n − 2 k > 0 , n − 2 k = 6 p − 1 , 6 p − 2 . This theorem can b e prov ed without big difficulties if one notes that b oth sides of equalit y (6 ) satisfy the same recurrence relatio n a s the n umbers in the P ascal triangle, (7) LH S ( n, k ) = LH S ( n − 1 , k ) + LH S ( n − 1 , k − 1) , RH S ( n, k ) = RH S ( n − 1 , k ) + R H S ( n − 1 , k − 1) , and the same initia l conditions (for k = 0 and k = n ). More pr ecisely , equalities (7) hold ev erywhere outside the v ertical line n − 2 k = 0 . On this line one should add 1 to the righ t hand sides of b oth equalities. Reference s [1] G. E. Andrews, T he theory of partitions, E ncyclop edia o f Mathematics and Its Applications, vol. 2, Addison-W esley , Reading , Mass.; reissue d b y Ca m bridge Univ ersity Pres s, Cambridge, 1985 . [2] I. S. Sominskii, Elementary a lgebra: a supplementary course, Fizmatlit, Moscow, 196 3 (in Russ ian). E-mail addr ess : stoyan@mcc me.ru