cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A256418 Congrua (possible solutions to the congruum problem): numbers k such that there are integers x, y and z with k = x^2-y^2 = z^2-x^2.

Original entry on oeis.org

24, 96, 120, 216, 240, 336, 384, 480, 600, 720, 840, 864, 960, 1080, 1176, 1320, 1344, 1536, 1920, 1944, 2016, 2160, 2184, 2400, 2520, 2880, 2904, 3000, 3024, 3360, 3456, 3696, 3840, 3960, 4056, 4320, 4704, 4896, 5280, 5376, 5400, 5544
Offset: 1

Views

Author

N. J. A. Sloane, Apr 06 2015, following a suggestion from Robert Israel, Apr 03 2015

Keywords

Comments

k is a "congruum" iff k/4 is the area of a Pythagorean triangle, so these are the numbers 4*A009112.
Each congruum is a multiple of 24; it cannot be a square.
This entry incorporates many comments that were originally in A057102. A057103 and A055096 need to be checked.

Examples

			a(11)=840 since 840=29^2-1^2=41^2-29^2 (indeed also 840=37^2-23^2=47^2-37^2).

Links

Crossrefs

Cf. A004431 for possible values of x in definition. Cf. A057103, A055096 for triangles of all congrua and values of x.
Cf. A009112, A073120, A135789, A135786.

Programs

  • Mathematica
    r[n_] := Reduce[0 < y < x && 0 < x < z && n == x^2 - y^2 == z^2 - x^2, {x, y, z}, Integers];
Reap[For[n = 24, n < 10^4, n += 24, rn = r[n]; If[rn =!= False, Print[n, " ", rn]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 25 2019 *)

A056203 Triangle of numbers related to congruum problem: T(n,k)=n^2+2nk-k^2 with n>k>0, starting at T(2,1)=7.

Original entry on oeis.org

7, 14, 17, 23, 28, 31, 34, 41, 46, 49, 47, 56, 63, 68, 71, 62, 73, 82, 89, 94, 97, 79, 92, 103, 112, 119, 124, 127, 98, 113, 126, 137, 146, 153, 158, 161, 119, 136, 151, 164, 175, 184, 191, 196, 199, 142, 161, 178, 193, 206, 217, 226, 233, 238, 241, 167, 188
Offset: 1

Views

Author

Henry Bottomley, Aug 02 2000

Keywords

Comments

The congruum problem is to find h (A057103) such that there are integers x (A055096), y (A057105) and z (A056203) with h=x^2-y^2=z^2-x^2.
Refers to A057102, which had an incorrect description and has been replaced by A256418. As a result the present sequence should be re-checked. - N. J. A. Sloane, Apr 06 2015

Examples

			a(1) = T(2,1) = 2^2+2*2*1-1 = 7.

Links

Crossrefs

Cf. A057102.

Formula

a(n) = sqrt(A057103(n)+A055096(n)^2) = sqrt(2*A057103(n)+A057105(n)^2).

A057105 Triangle of numbers (when unsigned) related to congruum problem: T(n,k)=k^2+2nk-n^2 with n>k>0 and starting at T(2,1)=1.

Original entry on oeis.org

1, -2, 7, -7, 4, 17, -14, -1, 14, 31, -23, -8, 9, 28, 49, -34, -17, 2, 23, 46, 71, -47, -28, -7, 16, 41, 68, 97, -62, -41, -18, 7, 34, 63, 94, 127, -79, -56, -31, -4, 25, 56, 89, 124, 161, -98, -73, -46, -17, 14, 47, 82, 119, 158, 199, -119, -92, -63, -32, 1, 36, 73, 112, 153, 196, 241, -142, -113, -82, -49, -14, 23, 62, 103
Offset: 1

Views

Author

Henry Bottomley, Aug 02 2000

Keywords

Comments

Signed values are only relevant for the explicit formula.
Refers to A057102, which had an incorrect description and has been replaced by A256418. As a result the present sequence should be re-checked. - N. J. A. Sloane, Apr 06 2015

Examples

			a(1)=T(2,1)=1^2+2*2*1-2^2=1

Links

Crossrefs

Cf. A057102. The congruum problem is about finding solutions for h (A057103) where there are integers x (A055096), y (A057105 unsigned) and z (A056203) such that h=x^2-y^2=z^2-x^2.

Formula

Unsigned: a(n) =sqrt(A055096(n)^2-A057103(n)) =sqrt(A056203(n)^2-2*A057103(n)).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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