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 1-4 of 4 results.

A085377 a(n) = 15n^2 + 13n^3.

Original entry on oeis.org

0, 28, 164, 486, 1072, 2000, 3348, 5194, 7616, 10692, 14500, 19118, 24624, 31096, 38612, 47250, 57088, 68204, 80676, 94582, 110000, 127008, 145684, 166106, 188352, 212500, 238628, 266814, 297136, 329672, 364500, 401698, 441344, 483516, 528292
Offset: 0

Views

Author

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Aug 12 2003

Keywords

Comments

Numbers that are the sum of three solutions of the Diophantine equation x^3 - y^3 = z^2.
Parametric representation of the solution is (x,y,z) = (8n^2, 7n^2, 13n^3), thus getting a(n) = 8n^2 + 7n^2 + 13n^3 = 15n^2 + 13n^3.
Geometrically, 13^2 = 8^3 - 7^3 means that the square of the hypotenuse of a Pythagorean triangle (5,12,13) is the difference of two cubes, which I recently found on p70 of David Wells' book "The Penguin Dictionary of Curios and Interesting Numbers", Penguin Books, 1997. See also A085479.

Crossrefs

Cf. A085409.

Programs

  • Mathematica
    Table[15n^2 + 13n^3, {n, 1, 34}]

Formula

a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). G.f.: 2*x*(14+26*x-x^2)/(1-x)^4. [From R. J. Mathar, Apr 20 2009]

Extensions

More terms from Robert G. Wilson v, Aug 16 2003
Edited by N. J. A. Sloane, Apr 29 2008

A085482 Product of three solutions of the Diophantine equation x^2 - y^2 = z^3.

Original entry on oeis.org

54, 13824, 354294, 3538944, 21093750, 90699264, 311299254, 905969664, 2324522934, 5400000000, 11575379574, 23219011584, 44049458934, 79692609024, 138396093750, 231928233984, 376690901814, 595077871104, 917112404214
Offset: 1

Views

Author

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Aug 15 2003

Keywords

Comments

Parametric representation of the solution is (x,y,z) = (6n^3, 3n^3, 3n^2), thus getting a(n) = 54*n^8.

Links

Crossrefs

Cf. A085409.

Programs

  • Maple
    A085482:=n->54*n^8; seq(A085482(n), n=1..50); # Wesley Ivan Hurt, Nov 26 2013
  • Mathematica
    54*Range[20]^8 (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{54,13824,354294,3538944,21093750,90699264,311299254,905969664,2324522934},20] (* Harvey P. Dale, Jul 10 2013 *)

Formula

a(n) = 54*n^8.
a(1)=54, a(2)=13824, a(3)=354294, a(4)=3538944, a(5)=21093750, a(6)=90699264, a(7)=311299254, a(8)=905969664, a(9)=2324522934, a(n)=9*a(n-1)- 36*a(n-2)+ 84*a(n-3)- 126*a(n-4)+126*a(n-5)- 84*a(n-6)+ 36*a(n-7)-9*a(n-8)+a(n-9). - Harvey P. Dale, Jul 10 2013

Extensions

More terms from Ray Chandler, Nov 06 2003

A089207 a(n) = 4n^3 + 2n^2.

Original entry on oeis.org

6, 40, 126, 288, 550, 936, 1470, 2176, 3078, 4200, 5566, 7200, 9126, 11368, 13950, 16896, 20230, 23976, 28158, 32800, 37926, 43560, 49726, 56448, 63750, 71656, 80190, 89376, 99238, 109800, 121086, 133120, 145926, 159528, 173950, 189216
Offset: 1

Views

Author

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Dec 09 2003

Keywords

Comments

Yet another parametric representation of the solutions of the Diophantine equation x^2 - y^2 = z^3 is (3n^3, n^3, 2n^2). By taking the sum x+y+z we get a(n) = 4n^3 + 2n^2.
If Y is a 3-subset of an 2n-set X then, for n>=5, a(n-2) is the number of 5-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 16 2007

Links

Crossrefs

Cf. A085409, A087887.

Programs

  • Mathematica
    Table[4n^3+2n^2,{n,40}] (* Harvey P. Dale, Jun 12 2020 *)

Formula

a(n) = 2*A099721(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). G.f.: 2*x*(3+8*x+x^2)/(x-1)^4. [R. J. Mathar, Apr 20 2009]
a(n) = 2 * n * A014105(n). - Richard R. Forberg, Jun 16 2013

Extensions

More terms from Ray Chandler, Feb 15 2004

A087887 a(n) = 18n^3 + 6n^2.

Original entry on oeis.org

0, 24, 168, 540, 1248, 2400, 4104, 6468, 9600, 13608, 18600, 24684, 31968, 40560, 50568, 62100, 75264, 90168, 106920, 125628, 146400, 169344, 194568, 222180, 252288, 285000, 320424, 358668, 399840, 444048, 491400, 542004, 595968, 653400, 714408, 779100, 847584
Offset: 0

Views

Author

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Oct 13 2003

Keywords

Comments

Another parametric representation of the solutions of the Diophantine equation x^2 - y^2 = z^3 is (x,y,z) = (15n^3, 3n^3, 6n^2), thus getting a(n) = 18n^3 + 6n^2.

Crossrefs

Cf. A036659, A085409, A085482.

Programs

  • Mathematica
    a[n_] := 18*n^3 + 6*n^2; Array[a, 50, 0] (* Amiram Eldar, Jan 10 2023 *)

Formula

O.g.f.: 12x(2+6x+x^2)/(-1+x)^4. a(n) = 12*A036659(n). - R. J. Mathar, Apr 07 2008
From Amiram Eldar, Jan 10 2023: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/36 + sqrt(3)*Pi/12 + 3*log(3)/4 - 3/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/72 - sqrt(3)*Pi/6 - log(2) + 3/2. (End)

Extensions

More terms from Ray Chandler, Nov 06 2003
Showing 1-4 of 4 results.

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