This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A346263 #25 Jan 06 2024 14:32:03 %S A346263 9,9,9,9,25,9,9,9,9,25,9,9,9,49,9,9,25,9,9,9,9,25,9,9,9,9,25,9,9,49,9, %T A346263 81,9,9,25,9,9,9,9,25,9,9,9,9,25,9,49,9,9,9,9,25,9,9,9,9,25,9,121,9,9, %U A346263 49,9,25,9,9,81,9,9,9,25,9,9,9,9,25,9,9,49,9,9 %N A346263 Irregular triangle of numbers T(n,k) read by rows where each T(n,k) is the number of large or small squares that are used to tile elementary squares of type 2 whose length of side is A344332(n). %C A346263 An elementary square of type 2 is the smallest square that can be tiled with squares of two different sides a < b satisfying a^2+b^2 = c^2 and so that the numbers of small and large squares are equal. %C A346263 Every term is an odd square >= 9 and each odd square is present infinitely many times. %C A346263 Notation: s_p (resp. s_e) = side of a primitive (resp. elementary) tiled square, a = side of small squares and b = side of large squares used to tile a primitive square, and z_p (z_e) = number of small squares = number of large squares used to tile a primitive (resp. elementary) square. %C A346263 A primitive square with side s_p = a*c/(c-b) is tiled with z_p small and z_p large squares with sides a and b, and z_p = (a/(c-b))^2. %C A346263 Each elementary square with a side s_e = k*s_p, k>0, is tiled with z_e small and z_e large squares with sides k*a and k*b, and z_e = z_p = (a/(c-b))^2. %C A346263 When an elementary side A344332(n) is a multiple of m distinct primitive sides s_p, then there are m different values T(n,1), ..., T(n,m) in the row n (see example). %e A346263 The triangle T begins: %e A346263 n\k 1 2 3 4 5 %e A346263 1: 9 %e A346263 2: 9 %e A346263 3: 9 %e A346263 4: 9 %e A346263 5: 25 %e A346263 6: 9 %e A346263 7: 9 %e A346263 8: 9 %e A346263 9: 9 %e A346263 10: 25 %e A346263 11: 9 %e A346263 12: 9 %e A346263 13: 9 %e A346263 14: 49 %e A346263 15: 9 %e A346263 16: 9, 25 %e A346263 17: 9 %e A346263 ... %e A346263 The first elementary side that is a multiple of two primitive sides (15 and 65) is A344332(16) = 195 = 13*15 = 3*65. %e A346263 As 195 = 13*15, the number z of squares with side a = 13*3 = 39 and b = 13*4 = 52 to tile this elementary square is T(16,1) = (39/(65-52))^2 = 9. %e A346263 As 195 = 3 * 65, the number z of squares with side a = 3*5 = 15 and b = 3*12 = 36 to tile this elementary square is T(16,2) = (15/(39-36))^2 = 25. %e A346263 Hence, the elementary square with side A344332(16) = 195 has two different possible tilings: with T(16,1) = 9 squares of sides (a,b) = (39,52) or with T(16,2) = 25 squares of sides (a,b) = (15,36). %e A346263 Elementary square 195 X 195 with a = 39, b = 52, s = 195, z = 9: %e A346263 ________ ________ ________ _____ %e A346263 | | | | | %e A346263 | | | | | %e A346263 | | | |_____| %e A346263 |________|________|________| | %e A346263 | | | | | %e A346263 | | | |_____| %e A346263 | | | | | %e A346263 |________|________|________| | %e A346263 | | | |_____| %e A346263 | | | | | %e A346263 | | | | | %e A346263 |_____ __|___ ____|_ ______|_____| %e A346263 | | | | | | %e A346263 | | | | | | %e A346263 |_____|______|______|______|_____| %Y A346263 Cf. A005917, A016754, A344330, A344332, A346264. %Y A346263 Cf. A345286 (similar for type 1). %K A346263 nonn,tabf %O A346263 1,1 %A A346263 _Bernard Schott_, Jul 13 2021