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 A346264 #46 Dec 25 2021 14:57:28
%S A346264 1,2,2,3,1,2,4,2,4,2,3,4,2,1,6,4,4,4,5,2,3,6,2,6,4,2,4,2,2,8,1,3,8,4,
%T A346264 6,2,8,2,2,6,4,4,4,6,9,2,4,7,8,2,8,2,4,6,1,6,4,3,2,2,10,3,2,6,4,12,2,
%U A346264 8,4,8,2,4,4,2,2,12,4,2,4,6,7,8,8,2,6,4,5,12,2,12,2,3,3,4
%N A346264 a(n) is the number of distinct possible tilings of type 2 for squares with side = A344332(n) and that can be tiled with squares of two different sizes so that the numbers of large or small squares are equal.
%C A346264 Every side of square of type 2 in A344332 is also the side of an elementary square of type 2. 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 A346264 Some notation: s = side of the tiled square, a = side of small squares, b = side of large squares, and z = number of small squares = number of large squares.
%C A346264 a(n) = 1 iff A344332(n) is a term of A005917 that is not a multiple of another term of A005917 (15, 65, 175, 369, 671, 2465, ...).
%C A346264 The first side that is a multiple of two primitive sides is 195 = 13*15 = 3*65 (see 3rd example).
%D A346264 Ivan Yashchenko, Invitation to a Mathematical Festival, pp. 10 and 102, MSRI, Mathematical Circles Library, 2013.
%H A346264 Bernard Schott, <a href="/A346264/a346264.txt">Tilings of type 2 for square 30 X 30 with a(2) = 2.</a>
%F A346264 a(n) = Sum_{(k>=2) & (A005917(k)|A344332(n))} tau(A344332(n)/A005917(k)).
%e A346264 ---> For a(1), A344332(1) = 15, then, with the formula, we get a(1) = tau(A344332(1)/A005917(2)) = tau(15/15) = tau(1) = 1, and the corresponding tiling of this smallest square 15 X 15 of type 2 consists of z = 9 squares whose sides (a,b) = (3,4) (see below).
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%e A346264 a(1) = 1
%e A346264 ---> For a(2), A344332(2) = 30, then, with the formula, we get a(2) = tau(A344332(2)/A005917(2)) = tau(30/15) = tau(2) = 2, and these 2 distinct tilings are:
%e A346264 1) 30 = 2*A344332(1) = 2*15, z(30) = 2^2 * z(15) = 4*9 = 36 and square 30 X 30 can be tiled with z = 36 squares whose sides (a,b) = (3,4), that is 4 copies of the elementary and primitive square 15 X 15 (as above). Also,
%e A346264 2) 30 = 1*A344332(2) = 1*30, z(30) = 1^2 * z(15) = 1*9 = 9 and the elementary square 30 X 30 can be tiled with z = 9 squares whose sides (a,b) = (6,8) (see link with corresponding drawings).
%e A346264 ---> For a(16), A344332(16) = 195, then, with the formula, we get a(16) = tau(A344333(16)/A005917(2)) + tau(A344333(16)/A005917(3)) = tau(195/15) + tau(195/65) = tau(13) + tau(3) = 2+2 = 4, and these 4 distinct tilings are:
%e A346264 1) 195 = 13*A344332(1) = 13*15, z_1(195) = 13^2 * z(15) = 169*9 = 1521 and square 195 X 195 can be tiled with z = 1521 squares whose sides (a,b) = (3,4), that is 169 copies of the elementary and primitive square 15 X 15, as above;
%e A346264 2) 195 = 1*A344332(16) = 1*195, z_2(195) = 1^2 * z(195) = 1*9 = 9 and the elementary square 195 X 195 can be tiled with z = 9 squares whose sides (a,b) = (39,52);
%e A346264 3) 195 = 3*A344332(5) = 3*65, z_3(195) = 3^2 * z(65) = 9*25 = 225 [z(65) = A346263(5) = T(5,1) = 25] and square 195 X 195 can be tiled with z = 225 squares whose sides (a,b) = (5,12), that is 9 copies of the elementary and primitive square 65 X 65;
%e A346264 4) 195 = 1*A344332(16) = 1*195, z_4(195) = 1^2 * z(195) = 1*25 = 25 and the elementary square 195 X 195 can be tiled with z = 25 squares whose sides (a,b) = (15,36).
%o A346264 (PARI) \\ using isok2 from A344332; isok7 is for A005917
%o A346264 isok7(k) = my(kk= sqrtnint(k\4, 3)+2); vecsearch(vector(kk, i, (i+1)^4 - i^4), k);
%o A346264 sd(x) = sumdiv(x, d, if (isok7(d), numdiv(x/d)));
%o A346264 lista(nn) = my(v2 = select(isok2, [1..nn])); apply(sd, v2);
%Y A346264 Cf. A005917, A016754, A344332, A346263.
%K A346264 nonn
%O A346264 1,2
%A A346264 _Bernard Schott_, Aug 09 2021