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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).
1
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, 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, 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
OFFSET
1,3
COMMENTS
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.
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.
REFERENCES
Ivan Yashchenko, Invitation to a Mathematical Festival, pp. 10 and 102, MSRI, Mathematical Circles Library, 2013.
FORMULA
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).
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).
If A344330(n) = A344331(k_1) = A344332(k_2) then a(n) = A345287(k_1) + A346264(k_2).
EXAMPLE
-> 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):
Primitive square 10 X 10 corresponding to a(1) = 1 with
a = 1, b = 2, s = 10, z = 20:
___ ___ _ ___ ___ _
| | |_| | |_|
|___|___|_|___|___|_|
| | |_| | |_|
|___|___|_|___|___|_|
| | |_| | |_|
|___|___|_|___|___|_|
| | |_| | |_|
|___|___|_|___|___|_|
| | |_| | |_|
|___|___|_|___|___|_|
-> 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):
Primitive square 15 X 15 corresponding to a(2) = 1 with
a = 3, b = 4, c = 5, s = 15, z = 9:
________ ________ ________ ______
| | | | |
| | | | |
| | | |______|
|_______ |________|________| |
| | | | |
| | | |______|
| | | | |
|________|________|________| |
| | | |______|
| | | | |
| | | | |
|_____ __|___ ____|_ ______|______|
| | | | | |
| | | | | |
|_____|______|______|______|______|
-> 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).
-> 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.
PROG
(PARI) \\ isok1 from A344331 and isok2 from A344332
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
sd7(x) = sumdiv(x, d, if (isok3(d), numdiv(x/d))); \\ A345287
isok7(k) = my(kk= sqrtnint(k\4, 3)+2); vecsearch(vector(kk, i, (i+1)^4 - i^4), k); \\ A005917
sd4(x) = sumdiv(x, d, if (isok7(d), numdiv(x/d))); \\ A346264
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Bernard Schott, Aug 11 2021
EXTENSIONS
a(19),a(59),a(86),a(87) corrected by Bernard Schott, Dec 23 2021
STATUS
approved