A374602 - OEIS

A374602

Array of successive integer solutions to sqrt((d-c)*b^2 + c*(b+1)^2) for nonsquare integers d >= 2 (d=A000037(n) for n >= 1), where b and c are positive integers and c < d, read by antidiagonals.

%I #31 Nov 15 2024 23:31:14

%S 5,29,3,169,11,5,985,41,13,3,5741,153,34,7,4,33461,571,89,18,5,10,

%T 195025,2131,233,29,11,11,4,1136689,7953,610,69,28,23,5,7,6625109,

%U 29681,1597,178,62,58,13,8,6,38613965,110771,4181,287,79,338,14,13,22,4

%N Array of successive integer solutions to sqrt((d-c)*b^2 + c*(b+1)^2) for nonsquare integers d >= 2 (d=A000037(n) for n >= 1), where b and c are positive integers and c < d, read by antidiagonals.

%C T(n,k) is the diagonal lengths of increasingly nearly regular d-dimensional Pythagorean hyperrectangles.

%C 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.

%C 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.

%C Square d produce solutions following a different pattern, shown as A375336.

%F T(n, 1) = A373666(A000037(n)).

%e n=row index; d=nonsquare integer of index n (A000037(n)):

%e n d T(n,k)

%e ---+----+-------------------------------------------------------------

%e 1 | 2 | 5, 29, 169, 985, 5741, 33461, 195025, 1136689, 6625109, ...

%e 2 | 3 | 3, 11, 41, 153, 571, 2131, 7953, 29681, 110771, ...

%e 3 | 5 | 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, ...

%e 4 | 6 | 3, 7, 18, 29, 69, 178, 287, 683, 1762, ...

%e 5 | 7 | 4, 5, 11, 28, 62, 79, 175, 446, 988, ...

%e 6 | 8 | 10, 11, 23, 58, 338, 373, 781, 1970, 11482, ...

%e 7 | 10 | 4, 5, 13, 14, 25, 62, 111, 148, 185, ...

%e 8 | 11 | 7, 8, 13, 32, 57, 139, 158, 259, 638, ...

%e 9 | 12 | 6, 22, 39, 69, 82, 125, 306, 543, 1142, ...

%e 10 | 13 | 4, 5, 7, 17, 30, 43, 53, 76, 185, ...

%e 11 | 14 | 9, 11, 14, 19, 46, 81, 267, 329, 418, ...

%e 12 | 15 | 6, 10, 21, 23, 30, 39, 94, 165, 362, ...

%e 13 | 17 | 25, 27, 34, 41, 98, 171, 260, 1649, 1779, ...

%e 14 | 18 | 6, 13, 15, 18, 21, 50, 87, 132, 198, ...

%e 15 | 19 | 5, 7, 8, 9, 11, 31, 34, 37, 56, ...

%e 16 | 20 | 10, 26, 68, 125, 159, 178, 197, 466, 807, ...

%e 17 | 21 | 6, 9, 12, 13, 14, 33, 57, 86, 134, ...

%e 18 | 22 | 5, 7, 8, 17, 18, 19, 31, 64, 77, ...

%e 19 | 23 | 16, 19, 27, 28, 29, 68, 117, 176, 764, ...

%e 20 | 24 | 6, 9, 11, 14, 36, 39, 57, 58, 59, ...

%e ...

%e sqrt((2-1)*1^2 + 1*(1+1)^2) = sqrt(5) -> not an integer so not included.

%e sqrt((2-1)*3^2 + 1*(3+1)^2) = 5 -> T(1,1).

%e sqrt((2-1)*20^2 + 1*(20+1)^2) = 29 -> T(1,2).

%e sqrt((3-2)*1^2 + 2*(1+1)^2) = 3 -> T(2,1).

%e sqrt((6-2)*7^2 + 2*(7+1)^2) = 18 -> T(4,3).

%o (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)

%o for(n=1, 20, print(n, " ", row(n, 10)))

%Y Row 1 is A001653 starting at n=2.

%Y Row 2 is A079935 starting at n=2.

%Y Bisection of row 2 starting with the first term is A189356 starting at n=1.

%Y Bisection of row 2 starting with the second term is A122769 starting at n=2.

%Y Row 3 is A001519 starting at n=3.

%Y Bisection of row 3 starting with the first term is A033889 starting at n=1.

%Y Bisection of row 3 starting with the second term is A033891 starting at n=1.

%Y Row 4 is A131093 starting at n=3.

%Y Cf. A000037, A373666.

%Y Cf. A377290, A377291, A375336.

%K nonn,tabl

%O 1,1

%A _Charles L. Hohn_, Jul 13 2024