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.

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.

A153642 a(n) = 4*n^2 + 24*n + 8.

Original entry on oeis.org

36, 72, 116, 168, 228, 296, 372, 456, 548, 648, 756, 872, 996, 1128, 1268, 1416, 1572, 1736, 1908, 2088, 2276, 2472, 2676, 2888, 3108, 3336, 3572, 3816, 4068, 4328, 4596, 4872, 5156, 5448, 5748, 6056, 6372, 6696, 7028, 7368, 7716, 8072, 8436, 8808, 9188
Offset: 1

Views

Author

Vincenzo Librandi, Dec 30 2008

Keywords

Comments

2*(fifth subdiagonal of triangle A144562).
Sequence gives values x of solutions (x, y) to the Diophantine equation x^3+28*x^2 = y^2. For a more comprehensive list of solutions see A155135.
For n >= 3, a(n - 1) is the number of checkmate positions with white queen and white king against black king on an n X n board. Reason: The black king can only be on the edge. There are 4*(4*n + 1) checkmate positions where the black king is in the corner, 4*(2*n + 4) checkmate positions where the black king is immediately adjacent to the corner square, and there are 4*(n - 4)*(n + 2) checkmate positions where the black king is on another edge square. That's a total of 4*n^2 + 16*n - 12 = a(n - 1) checkmate positions. - Felix Huber, Oct 29 2023

Links

Crossrefs

Cf. A028881, A144562, A155135, A155136,

Programs

Formula

a(n) = A155135(2n+8) = A155136(2n+7).
a(n) = 4*A028881(n+3).
G.f.: 4*(3 - x)*(3 - 2*x)/(1-x)^3.
a(n)= 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: 4*(-2 + (2 + 7*x + x^2)*exp(x)). - G. C. Greubel, Aug 23 2016
From Amiram Eldar, Mar 02 2023: (Start)
Sum_{n>=1} 1/a(n) = 1/56 - cot(sqrt(7)*Pi)*Pi/(8*sqrt(7)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 31/168 - cosec(sqrt(7)*Pi)*Pi/(8*sqrt(7)). (End)

Extensions

Edited and extended by Klaus Brockhaus, Jan 21 2009

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)
Showing 1-3 of 3 results.

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