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.

A155135 Integers n such that n^3+28*n^2 is a square.

Original entry on oeis.org

-28, -27, -24, -19, -12, -3, 0, 8, 21, 36, 53, 72, 93, 116, 141, 168, 197, 228, 261, 296, 333, 372, 413, 456, 501, 548, 597, 648, 701, 756, 813, 872, 933, 996, 1061, 1128, 1197, 1268, 1341, 1416, 1493, 1572, 1653, 1736, 1821, 1908, 1997, 2088, 2181, 2276, 2373
Offset: 1

Views

Author

Klaus Brockhaus, Jan 21 2009

Keywords

Comments

Values x of solutions (x, y) to the Diophantine equation x^3+28*x^2 = y^2. Corresponding values y are in A155137.
Agrees with A155136 except for insertion of zero after a(6) = 3.

Examples

			For n = -19, n^3+28*n^2 = -6859+10108 = 3249 = 57^2 is a square.
For n = 0, n^3+28*n^2 = 0^3+28*0^2 = 0 = 0^2 is a square.
For n = 21; n^3+28*n^2 = 9261+12348 = 21609 = 147^2 is a square.

Links

Crossrefs

Cf. A155136, A155137, A153642.

Programs

  • Magma
    [ n: n in [ -30..2400] | IsSquare(n^3+28*n^2) ];
  • Mathematica
    CoefficientList[Series[-(28-57*x+27*x^2+8*x^6-11*x^7+3*x^9)/(1-x)^3,{x,0,60}],x] (* Vincenzo Librandi, Feb 22 2012 *)
Select[Range[-30,2500],IntegerQ[Sqrt[#^3+28#^2]]&] (* or *) LinearRecurrence[ {3,-3,1},{-28,-27,-24,-19,-12,-3,0,8,21,36},60] (* Harvey P. Dale, Jan 10 2023 *)

Formula

G.f.: -(28-57*x+27*x^2+8*x^6-11*x^7+3*x^9)/(1-x)^3.

A155136 Integers k such that k + 28 is a square.

Original entry on oeis.org

-28, -27, -24, -19, -12, -3, 8, 21, 36, 53, 72, 93, 116, 141, 168, 197, 228, 261, 296, 333, 372, 413, 456, 501, 548, 597, 648, 701, 756, 813, 872, 933, 996, 1061, 1128, 1197, 1268, 1341, 1416, 1493, 1572, 1653, 1736, 1821, 1908, 1997, 2088, 2181, 2276, 2373
Offset: 1

Views

Author

Klaus Brockhaus, Jan 21 2009

Keywords

Comments

Values x of nonzero solutions (x,y) to the Diophantine equation x^3 + 28*x^2 = y^2. Corresponding values y are in A155137.
Agrees with A155135 except for omission of zero after a(6) = 3.

Examples

			For k = -19, k + 28 = 9 = 3^2 is a square.
For k = -3, k + 28 = 25 = 5^2 is a square.
For k = 21, k + 28 = 49 = 7^2 is a square.

Links

Crossrefs

Cf. A153642, A155135, A155137, A155138.

Programs

  • Magma
    [ n: n in [ -30..2500] | IsSquare(n+28) ];
  • Mathematica
    Range[0,50]^2-28 (* or  *) LinearRecurrence[{3,-3,1},{-28,-27,-24},50] (* Harvey P. Dale, May 15 2023 *)
  • PARI
    a(n)=n^2-2*n-27 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = n^2 - 2*n - 27.
G.f.: -(4-3*x)*(7-9*x)/(1-x)^3.
From Elmo R. Oliveira, Oct 31 2024: (Start)
E.g.f.: exp(x)*(x^2 - x - 27) + 27.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A155137 a(n) = nonnegative value y such that (A155135(n), y) is a solution to the Diophantine equation x^3+28*x^2 = y^2.

Original entry on oeis.org

0, 27, 48, 57, 48, 15, 0, 48, 147, 288, 477, 720, 1023, 1392, 1833, 2352, 2955, 3648, 4437, 5328, 6327, 7440, 8673, 10032, 11523, 13152, 14925, 16848, 18927, 21168, 23577, 26160, 28923, 31872, 35013, 38352, 41895, 45648, 49617, 53808, 58227
Offset: 1

Views

Author

Klaus Brockhaus, Jan 21 2009

Keywords

Comments

Agrees with A155138 except for insertion of zero after a(6) = 15.

Examples

			(A155135(3), a(3)) = (-24, 48) is a solution: (-24)^3+28*(-24)^2 = -13824+16128 = 2304 = 48^2.
(A155135(7), a(8)) = (0, 0) is a solution: 0^3+28*0^2 = 0 = 0^2.
(A155135(8), a(8)) = (8, 48) is a solution: 8^3+28*8^2 = 512+1792 = 2304 = 48^2.

Links

Crossrefs

Cf. A155135, A155138, A153642.

Programs

  • Magma
    [ Integers()!SquareRoot(a) : n in [ -30..1500] | IsSquare(a) where a is n^3+28*n^2 ];
  • Mathematica
    CoefficientList[Series[3x (9-20x+9x^2+16x^5+x^6-19x^7+x^8+5x^9)/(1-x)^4,{x,0,40}],x] (* or *) LinearRecurrence[{4,-6,4,-1},{0,27,48,57,48,15,0,48,147,288,477},50] (* Harvey P. Dale, Sep 02 2021 *)
  • PARI
    a(n)=if(n>6, n^3 - 3*n^2 - 25*n + 27, [0, 27, 48, 57, 48, 15, 0][n+1]) \\ Charles R Greathouse IV, Oct 18 2022

Formula

G.f.: 3*x*(9-20*x+9*x^2+16*x^5+x^6-19*x^7+x^8+5*x^9)/(1-x)^4.

A155138 a(n) = nonnegative value y such that (A155136(n), y) is a solution to the Diophantine equation x^3+28*x^2 = y^2.

Original entry on oeis.org

0, 27, 48, 57, 48, 15, 48, 147, 288, 477, 720, 1023, 1392, 1833, 2352, 2955, 3648, 4437, 5328, 6327, 7440, 8673, 10032, 11523, 13152, 14925, 16848, 18927, 21168, 23577, 26160, 28923, 31872, 35013, 38352, 41895, 45648, 49617, 53808, 58227, 62880
Offset: 1

Views

Author

Klaus Brockhaus, Jan 21 2009

Keywords

Comments

Agrees with A155137 except for omission of zero after a(6) = 15.

Examples

			(A155136(4), a(4)) = (-19, 57) is a solution: (-19)^3+28*(-19)^2 = -6859+10108 = 3249 = 57^2.
(A155136(8), a(8)) = (21, 147) is a solution: 21^3+28*21^2 = 9261+12348 = 21609 = 147^2.

Crossrefs

Cf. A155136, A155137, A153642.

Programs

  • Magma
    [ Abs((n-1)^3-28*(n-1)): n in [1..41] ];
  • Mathematica
    Abs[#^3-28#]&/@Range[0,40] (* Harvey P. Dale, Aug 30 2016 *)

Formula

a(n) = Abs((n-1)^3-28*(n-1)).
G.f.: 3*x*(9-20*x+9*x^2+32*x^5-30*x^6-8*x^7+10*x^8)/(1-x)^4.
Showing 1-4 of 4 results.

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