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 A346265 #36 Dec 23 2021 17:33:22
%S A346265 1,1,2,5,3,2,2,9,1,1,2,2,1,4,9,4,2,2,13,5,3,2,4,10,2,5,2,2,1,16,2,4,6,
%T A346265 2,10,4,1,4,2,2,17,3,2,9,13,3,6,2,3,19,2,3,4,6,2,10,6,2,7,23,2,2,3,4,
%U A346265 18,8,4,4,2,18,2,2,6,2,18,2,4,2,4,2,2,21,3,4,6,11,14,6,2,23
%N A346265 a(n) is the number of distinct possible tilings of type 1 (A344331) or of type 2 (A344332) for a square whose side is A344330(n).
%C A346265 These squares with side = A344330(n) can be tiled with squares of two different sizes so that the numbers of large or small squares are equal.
%C A346265 Notation: s = side of the tiled squares, a = side of small squares, b = side of large squares, and z = number of small squares = number of large squares.
%D A346265 Ivan Yashchenko, Invitation to a Mathematical Festival, pp. 10 and 102, MSRI, Mathematical Circles Library, 2013.
%H A346265 Bernard Schott, <a href="/A346265/a346265_1.txt">The 5 distinct tilings corresponding to a(4) = 5</a>.
%F A346265 If A344330(n) = A344331(k_1) and there is no k_2 such that A344330(n) = A344332(k_2) then a(n) = A345287(k_1).
%F A346265 If A344330(n) = A344332(k_2) and there is no k_1 such that A344330(n) = A344331(k_1) then a(n) = A346264(k_2).
%F A346265 If A344330(n) = A344331(k_1) = A344332(k_2) then a(n) = A345287(k_1) + A346264(k_2).
%e A346265 -> A344330(1) = A344331(1) = 10 and there is no k_2 such that A344330(1) = A344332(k_2) = 10, then a(1) = A345287(1) = 1 (example below of type 1):
%e A346265 Primitive square 10 X 10 corresponding to a(1) = 1 with
%e A346265 a = 1, b = 2, s = 10, z = 20:
%e A346265 ___ ___ _ ___ ___ _
%e A346265 | | |_| | |_|
%e A346265 |___|___|_|___|___|_|
%e A346265 | | |_| | |_|
%e A346265 |___|___|_|___|___|_|
%e A346265 | | |_| | |_|
%e A346265 |___|___|_|___|___|_|
%e A346265 | | |_| | |_|
%e A346265 |___|___|_|___|___|_|
%e A346265 | | |_| | |_|
%e A346265 |___|___|_|___|___|_|
%e A346265 -> A344330(2) = A344332(1) = 15 and there is no k_1 such that A344330(2) = A344331(k_1) = 15, then a(2) = A346264(1) = 1 (example below of type 2):
%e A346265 Primitive square 15 X 15 corresponding to a(2) = 1 with
%e A346265 a = 3, b = 4, c = 5, s = 15, z = 9:
%e A346265 ________ ________ ________ ______
%e A346265 | | | | |
%e A346265 | | | | |
%e A346265 | | | |______|
%e A346265 |_______ |________|________| |
%e A346265 | | | | |
%e A346265 | | | |______|
%e A346265 | | | | |
%e A346265 |________|________|________| |
%e A346265 | | | |______|
%e A346265 | | | | |
%e A346265 | | | | |
%e A346265 |_____ __|___ ____|_ ______|______|
%e A346265 | | | | | |
%e A346265 | | | | | |
%e A346265 |_____|______|______|______|______|
%e A346265 -> A344330(4) = A344331(3) = A344332(2) = 30, then a(4) = A345287(3) + A346264(2) = 3+2 = 5 (see link with the corresponding 5 distinct tilings).
%e A346265 -> A344330(6) = A344332(3) = 45 and there is no k_1 such that A344330(6) = A344331(k_1) = 45, then a(6) = A346264(3) = 2.
%o A346265 (PARI) \\ isok1 from A344331 and isok2 from A344332
%o A346265 isok3(s) = {if (!(s % 2) && ispower(s/2, 4), return (0)); my(m = sqrtnint(s, 3)); vecsearch(setbinop((x, y)->if (gcd(x, y)==1, x*y*(x^2+y^2), 0), [1..m]), s); } \\ A344333
%o A346265 sd7(x) = sumdiv(x, d, if (isok3(d), numdiv(x/d))); \\ A345287
%o A346265 isok7(k) = my(kk= sqrtnint(k\4, 3)+2); vecsearch(vector(kk, i, (i+1)^4 - i^4), k); \\ A005917
%o A346265 sd4(x) = sumdiv(x, d, if (isok7(d), numdiv(x/d))); \\ A346264
%o A346265 lista(nn) = {for (n=1, nn, my(b1 = isok1(n), b2 = isok2(n)); if (b1 || b2, my(x = 0); if (b1, x += sd7(n)); if (b2, x += sd4(n)); print1(x, ", ");););} \\ _Michel Marcus_, Dec 23 2021
%Y A346265 Cf. A344330, A344331, A344332, A345287, A346264.
%K A346265 nonn
%O A346265 1,3
%A A346265 _Bernard Schott_, Aug 11 2021
%E A346265 a(19),a(59),a(86),a(87) corrected by _Bernard Schott_, Dec 23 2021