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.

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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Refers to A057102, which had an incorrect description and has been replaced by A256418.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)).

A135797 Numbers of the form x^4 + 6*x^2*y^2 + y^4 (where x,y are positive integers).

Original entry on oeis.org

8, 41, 128, 136, 313, 353, 648, 656, 776, 1201, 1241, 1513, 2048, 2056, 2176, 2696, 3281, 3321, 3593, 4481, 5000, 5008, 5128, 5648, 7048, 7321, 7361, 7633, 8521
Offset: 1

Views

Author

Artur Jasinski, Nov 29 2007

Keywords

Comments

Squares of these numbers are of the form N^4-M^2 (where N belongs to A135786 and M to A135796). Proof uses: (x^4+6*x^2*y^2+y^4)^2 = (x^2-y^2)^4+(4*x^3*y+4*x*y^3)^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Refers to A057102, which had an incorrect description and has been replaced by A256418.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

Crossrefs

Cf. A000404, A050803, A057102, A135784, A060803, A135786, A135787, A135789, A135790, A135791, A135792, A135793, A135795, A135796, A135796.

Programs

  • Mathematica
    a = {}; Do[Do[w = x^4 + 6x^2 y^2 + y^4; If[w < 10000, AppendTo[a, w]], {x, y, 1000}], {y, 1, 1000}]; Union[a]
(* alternate program *)
Union[Select[#[[1]]^4+6#[[1]]^2 #[[2]]^2+#[[2]]^4&/@Tuples[Range[ 1000],2], #<10000&]] (* Harvey P. Dale, Oct 07 2012 *)

A135794 Numbers of the form x^5 + 10*x^3*y^2 + 5*x*y^4 (where x,y are integers).

Original entry on oeis.org

16, 121, 122, 496, 512, 528, 1441, 1562, 1563, 1684, 3376, 3872, 3888, 3904, 4400, 6841, 8282, 8403, 8404, 8525, 9966, 12496, 15872, 16368, 16384, 16400, 16896, 20272, 21121, 27962, 29403, 29524, 29525, 29646, 31087, 33616, 37928, 46112
Offset: 1

Views

Author

Artur Jasinski, Nov 29 2007, Oct 10 2008

Keywords

Comments

Squares of these numbers are of the form N^5+M^2 (where N belongs to A000404 and M to A135795). Proof uses: (x^5+10x^3 y^2+5xy^4)^2=(x^2-y^2)^5+(5x^4y+10x^2y^3+y^5)^2.
Also numbers of the form ((y + x)^5 - (y - x)^5)/2 = x^5 + 10*x^3*y^2 + 5*x*y^4. - Artur Jasinski, Oct 10 2008
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

Crossrefs

Cf. A000404, A050803, A057102, A135784, A060803, A135786, A135787, A135789, A135790, A135791, A135792, A135793, A135795, A135796.

Programs

  • Mathematica
    a = {}; Do[Do[w = x^5 + 10x^3 y^2 + 5x y^4; If[w < 100000, AppendTo[a, w]], {x, 1, 1000}], {y, 1, 1000}]; Union[a]
Previous Showing 11-13 of 13 results.

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

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