A377290 -id:A377290 - cp's OEIS frontend

A377291 For each row n in array A374602(n, k), the asymptotic geometric growth factor of every A377290(n) terms, represented by its nearest integer.

6, 14, 7, 98, 16, 34, 1442, 398, 194, 119, 30, 62, 4354, 1154, 115598, 322, 23, 155234, 48, 98, 10402, 2702, 64514, 727, 482, 3040, 1154, 2114, 70, 142, 21314, 5474, 2498, 1442, 16793602, 674, 48497294, 158402, 47, 48670, 96, 194, 39202, 9998, 1684802, 2599

Offset: 1

Charles L. Hohn, Oct 23 2024

nonn

(a(n)^2-4)/A000037(n) is a square, and as such, a(n) is a member of row x of A298675(x, k), where x is the smallest value >= 3 such that (x^2-4)/A000037(n) is a square. E.g. for n=38: A000037(38)=44, x=20 ((20^2-4)/44 = 3^2), and a(38) = 158402 = A298675(20, 4).

The same row x of A298675(x, k) also results as integer solutions to g+(1/g) where g=(w*sqrt(d) + ceiling(w*sqrt(d)))/2 and d=A000037(n) for integers w >= 1. As such, it follows that g(n) can be expressed as a simple integer arithmetic transformation of sqrt(A000037(n)), e.g. g(1) = 2*sqrt(2)+3 (A156035), g(2) = 4*sqrt(3)+7 (A354129), g(3) = (3*sqrt(5)+7)/2 (A374883), g(4) = 20*sqrt(6)+49, and g(5) = 3*sqrt(7)+8 (A010516+8).

			For n = 5, the first few terms of A374602(5, k) are {4, 5, 11, 28, 62, 79, 175, 446, 988} and the period size is A377290(5) = 4, giving A374602(5, 1+4)/A374602(5, 1) = 62/4 = 15.5, 79/5 = 15.8, 175/11 = 15.909..., 446/28 = 15.928..., 988/62 = 15.935..., ..., to limit 15.937... -> g(5), from which g(5)+(1/g(5)) = 16 -> a(5).

Charles L. Hohn, Table of n, a(n) for n = 1..90

Cf. A374602, A377290, A000037, A298675.

Cf. A156035, A354129, A374883, A010516.

Growth factor g(n) = Lim_{k->oo}(A374602(n, k+A377290(n))/A374602(n, k)).
a(n) = round(g(n)) = ceiling(g(n)) = g(n)+(1/g(n)).
Inverse: g(n) = (sqrt(a(n)^2-4)+a(n))/2.
For d = A000037(n) and x in {1, 2, 4}, when d+x is a square (unless x==4 and d+x is even): a(n) = 4/x*d+2.
For d = A000037(n) and x in {-4, 1, 2, 4}, when n > 3 and d-x is a square (unless x==-4 and d-x is odd): a(n) = (4/abs(x))^2*d^2-16/x*d+2.

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.

Original entry on oeis.org

5, 29, 3, 169, 11, 5, 985, 41, 13, 3, 5741, 153, 34, 7, 4, 33461, 571, 89, 18, 5, 10, 195025, 2131, 233, 29, 11, 11, 4, 1136689, 7953, 610, 69, 28, 23, 5, 7, 6625109, 29681, 1597, 178, 62, 58, 13, 8, 6, 38613965, 110771, 4181, 287, 79, 338, 14, 13, 22, 4

Offset: 1

Charles L. Hohn, Jul 13 2024

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.

			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).

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.
Cf. A000037, A373666.
Cf. A377290, A377291, A375336.

row(n, c)=my(v=List(), d=n+floor(sqrt(n)+1/2) /* d=A000037(n) */, t=ceil(sqrt(d))); while(#v

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

cp's OEIS Frontend

A377291 For each row n in array A374602(n, k), the asymptotic geometric growth factor of every A377290(n) terms, represented by its nearest integer.

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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.

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cp's OEIS Frontend

A377291 For each row n in array A374602(n, k), the asymptotic geometric growth factor of every A377290(n) terms, represented by its nearest integer.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

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.