A155136 -id:A155136 - cp's OEIS frontend

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.

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

Klaus Brockhaus, Jan 21 2009

nonn

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

			(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.

Cf. A155136, A155137, A153642.

[ Abs((n-1)^3-28*(n-1)): n in [1..41] ];

Abs[#^3-28#]&/@Range[0,40] (* Harvey P. Dale, Aug 30 2016 *)

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.

A114962 a(n) = n^2 + 14.

Original entry on oeis.org

14, 15, 18, 23, 30, 39, 50, 63, 78, 95, 114, 135, 158, 183, 210, 239, 270, 303, 338, 375, 414, 455, 498, 543, 590, 639, 690, 743, 798, 855, 914, 975, 1038, 1103, 1170, 1239, 1310, 1383, 1458, 1535, 1614, 1695, 1778, 1863, 1950, 2039, 2130, 2223, 2318, 2415, 2514

Offset: 0

Cino Hilliard, Feb 21 2006

Old name was: "Numbers of the form x^2 + 14".

x^2 + 14 != y^n for all x,y and n > 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
J. H. E. Cohn, The diophantine equation x^2 + C = y^n, Acta Arithmetica LXV.4 (1993).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A155136, n^2 - 28; A000290, n^2; A114948, n^2 + 10.

Cf. sequences of the type n^2 + k: A002522 (k=1), A059100 (k=2), A117950 (k=3), A087475 (k=4), A117951 (k=5), A114949 (k=6), A117619 (k=7), A189833 (k=8), A189834 (k=9), A114948 (k=10), A189836 (k=11), A241748 (k=12), A241749 (k=13), this sequence (k=14), A241750 (k=15), A241751 (k=16), A241847 (k=17), A241848 (k=18), A241849 (k=19), A241850 (k=20), A241851 (k=21), A114963 (k=22), A241889 (k=23), A241890 (k=24), A114964 (k=30).

[n^2+14: n in [0..60]]; // Vincenzo Librandi, Apr 30 2014

Table[n^2 + 14, {n, 0, 50}]  (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)
CoefficientList[Series[(14 - 27 x + 15 x^2)/(1 - x)^3, {x, 0, 80}], x] (* Vincenzo Librandi, Apr 30 2014 *)

G.f.: (14-27*x+15*x^2)/(1-x)^3. - Colin Barker, Jan 11 2012
From Amiram Eldar, Nov 02 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(14)*Pi*coth(sqrt(14)*Pi))/28.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(14)*Pi*cosech(sqrt(14)*Pi))/28. (End)
From Elmo R. Oliveira, Nov 29 2024: (Start)
E.g.f.: exp(x)*(14 + x + x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

Added 14 from Vincenzo Librandi, Apr 30 2014
Definition changed by Bruno Berselli, Mar 13 2015
Offset corrected by Amiram Eldar, Nov 02 2020

A153642 a(n) = 4n^2 + 24n + 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

Vincenzo Librandi, Dec 30 2008

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A028881, A144562, A155135, A155136,

Magma
```
[ 4*(n+3)^2-28: n in [1..45] ];
```

LinearRecurrence[{3, -3, 1}, {36, 72, 116}, 50] (* Vincenzo Librandi, Feb 25 2012 *)

a(n)=4*n*(n+6)+8 \\ Charles R Greathouse IV, Dec 28 2011

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)

Edited and extended by Klaus Brockhaus, Jan 21 2009

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

Klaus Brockhaus, Jan 21 2009

sign

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.

			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.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A155136, A155137, A153642.

[ n: n in [ -30..2400] | IsSquare(n^3+28*n^2) ];

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 *)

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

cp's OEIS Frontend

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.

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A114962 a(n) = n^2 + 14.

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

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

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Examples

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Magma

Mathematica

Formula

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