id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-3952	Punpim, Jiramate; Jitman, Somphong	Characterizations and Identities for Isosceles Triangular Numbers	2021	16	.pdf	application/pdf	5924	336	79	− 2l2 − 2nl + 2l − 2l − 2 2 ) = l ( n2 + n2(n2 − 1)l 2 ) − (2n2 − 2n + 1) ( 2(l + l(l − 1)(n− 1) 2 )− l − 1 ) = lT (n2, l)− (2n2 − 2n + 1)(2T (l, n− 1)− (l + 1)). Then N = n + n(n− 1)l 2 for some positive integer n. It follows that rN + s = (2l + 1)2 ( n + n(n− 1)l 2 ) + l3 − 3l2 + 4 2 = n2l(2l + 1)2 + (2− l)n(2l + 1)2 + l3 − 3l2 + 4 2 = 4l3n2 + 4l2n2 + ln2 + 8l2n + 8ln + 2n− 4l3n− 4l2n− ln + l3 − 3l2 + 4 2 = (4l3 + 4l2 + l)n2 − (4l3 − 4l2 − 7l − 2)n + (l3 − 3l2 + 4) 2 = l ( (4l2 + 4l + 1)n2 − (4l2 − 6l − 4)n + (l2 − 4l + 4) ) − (l − 2) ((2l + 1)n− (l − 2)) 2 = l ((2l + 1)n− (l − 2))2 − (l − 2) ((2l + 1)n− (l − 2)) 2 = 2 ((2l + 1)n− (l − 2)) + l ((2l + 1)n− (l − 2))2 − l ((2l + 1)n− (l − 2)) 2 = ((2l + 1)n− (l − 2))	cache/ejpam-3952.pdf	txt/ejpam-3952.txt