OFFSET
1,1
COMMENTS
T(n,k) is the diagonal lengths of increasingly nearly regular d-dimensional Pythagorean hyperrectangles.
Each row n divides into equal length, geometrically periodic subsequences, each with its own subsequence period length (A377290) and geometric growth factor (A377291); it is conjectured that this is the case for all n, and that all solutions conform as such and that there are no solutions that do not, but these are not proven.
It is also not known if there is an algorithm for generating values for all rows other than testing all possible values for a row until a subsequence pattern emerges.
Square d produce solutions following a different pattern, shown as A375336.
EXAMPLE
n=row index; d=nonsquare integer of index n (A000037(n)):
n d T(n,k)
---+----+-------------------------------------------------------------
1 | 2 | 5, 29, 169, 985, 5741, 33461, 195025, 1136689, 6625109, ...
2 | 3 | 3, 11, 41, 153, 571, 2131, 7953, 29681, 110771, ...
3 | 5 | 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, ...
4 | 6 | 3, 7, 18, 29, 69, 178, 287, 683, 1762, ...
5 | 7 | 4, 5, 11, 28, 62, 79, 175, 446, 988, ...
6 | 8 | 10, 11, 23, 58, 338, 373, 781, 1970, 11482, ...
7 | 10 | 4, 5, 13, 14, 25, 62, 111, 148, 185, ...
8 | 11 | 7, 8, 13, 32, 57, 139, 158, 259, 638, ...
9 | 12 | 6, 22, 39, 69, 82, 125, 306, 543, 1142, ...
10 | 13 | 4, 5, 7, 17, 30, 43, 53, 76, 185, ...
11 | 14 | 9, 11, 14, 19, 46, 81, 267, 329, 418, ...
12 | 15 | 6, 10, 21, 23, 30, 39, 94, 165, 362, ...
13 | 17 | 25, 27, 34, 41, 98, 171, 260, 1649, 1779, ...
14 | 18 | 6, 13, 15, 18, 21, 50, 87, 132, 198, ...
15 | 19 | 5, 7, 8, 9, 11, 31, 34, 37, 56, ...
16 | 20 | 10, 26, 68, 125, 159, 178, 197, 466, 807, ...
17 | 21 | 6, 9, 12, 13, 14, 33, 57, 86, 134, ...
18 | 22 | 5, 7, 8, 17, 18, 19, 31, 64, 77, ...
19 | 23 | 16, 19, 27, 28, 29, 68, 117, 176, 764, ...
20 | 24 | 6, 9, 11, 14, 36, 39, 57, 58, 59, ...
...
sqrt((2-1)*1^2 + 1*(1+1)^2) = sqrt(5) -> not an integer so not included.
sqrt((2-1)*3^2 + 1*(3+1)^2) = 5 -> T(1,1).
sqrt((2-1)*20^2 + 1*(20+1)^2) = 29 -> T(1,2).
sqrt((3-2)*1^2 + 2*(1+1)^2) = 3 -> T(2,1).
sqrt((6-2)*7^2 + 2*(7+1)^2) = 18 -> T(4,3).
PROG
(PARI) row(n, c)=my(v=List(), d=n+floor(sqrt(n)+1/2) /* d=A000037(n) */, t=ceil(sqrt(d))); while(#v<c, my(b=floor(sqrt(t^2/d))); if ((t^2-d*b^2)%(b*2+1)==0, listput(v, t)); t++); concat(v)
for(n=1, 20, print(n, " ", row(n, 10)))
CROSSREFS
Row 1 is A001653 starting at n=2.
Row 2 is A079935 starting at n=2.
Bisection of row 2 starting with the first term is A189356 starting at n=1.
Bisection of row 2 starting with the second term is A122769 starting at n=2.
Row 3 is A001519 starting at n=3.
Bisection of row 3 starting with the first term is A033889 starting at n=1.
Bisection of row 3 starting with the second term is A033891 starting at n=1.
Row 4 is A131093 starting at n=3.
KEYWORD
AUTHOR
Charles L. Hohn, Jul 13 2024
STATUS
approved