A122769 -id:A122769 - cp's OEIS frontend

A122770 Numbers k such that A056109(k) is a square.

0, 6, 88, 1230, 17136, 238678, 3324360, 46302366, 644908768, 8982420390, 125108976696, 1742543253358, 24270496570320, 338044408731126, 4708351225665448, 65578872750585150, 913395867282526656, 12721963269204788038, 177194089901584505880

Offset: 0

Zak Seidov, Oct 21 2006

All terms are even. Sequence is infinite. Corresponding squares are s^2 with s = 1, 11, 153, 2131, 29681, 413403, 5757961, 80198051, 1117014753, 15558008491, 216695104121, 3018173449203, 42037733184721, ... (see A122769).
Numbers m such that the distance from (0,0,-1) to (m,m,m) in R^3 is an integer. - James R. Buddenhagen, Jun 15 2013
Also n such that the sum of the pentagonal numbers P(n) and P(n+1) is equal to the sum of two consecutive triangular numbers. - Colin Barker, Dec 07 2014

Colin Barker, Table of n, a(n) for n = 0..874
Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Cf. A001353, A001075, A011916, A056109, A122769, A251730.

LinearRecurrence[{15, -15, 1}, {0, 6, 88}, 25] (* Paolo Xausa, Jul 19 2024 *)

concat(0, Vec(2*x*(x-3) / ((x-1)*(x^2-14*x+1)) + O(x^100))) \\ Colin Barker, Dec 07 2014

a(n) = ((b+1)*(7+4*b)^n - (b-1)*(7-4*b)^n - 2)/6, where b = sqrt(3).
a(n) = 14*a(n-1) - a(n-2) + 4, with a(0)=0, a(1)=6.
a(n) = 2*A011916(n) = (A001353(n+1)^2 - A001075(n)^2)/2. - Richard R. Forberg, Aug 26 2013
a(n) = 15*a(n-1)-15*a(n-2)+a(n-3). - Colin Barker, Dec 07 2014
G.f.: 2*x*(x-3) / ((x-1)*(x^2-14*x+1)). - Colin Barker, Dec 07 2014

More terms from Colin Barker, Dec 07 2014

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)).

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A122770 Numbers k such that A056109(k) is a square.

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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

A122770 Numbers k such that A056109(k) is a square.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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.