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.

Previous Showing 11-18 of 18 results.

A219389 Numbers k such that 13*k+1 is a square.

Original entry on oeis.org

0, 11, 15, 48, 56, 111, 123, 200, 216, 315, 335, 456, 480, 623, 651, 816, 848, 1035, 1071, 1280, 1320, 1551, 1595, 1848, 1896, 2171, 2223, 2520, 2576, 2895, 2955, 3296, 3360, 3723, 3791, 4176, 4248, 4655, 4731, 5160, 5240, 5691, 5775, 6248, 6336, 6831, 6923
Offset: 1

Views

Author

Bruno Berselli, Nov 19 2012

Keywords

Comments

Equivalently, numbers of the form m*(13*m+2), where m = 0,-1,1,-2,2,-3,3,...
Also, integer values of h*(h+2)/13.

Links

Crossrefs

Cf. similar sequences listed in A219257.
Cf. A175886 (square roots of 13*a(n)+1).

Programs

  • Magma
    [n: n in [0..7000] | IsSquare(13*n+1)];
  • Magma
    I:=[0,11,15,48,56]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013
  • Maple
    A219389:=proc(q)
local n;
for n from 1 to q do if type(sqrt(13*n+1), integer) then print(n);
fi; od; end:
A219389(1000); # Paolo P. Lava, Feb 19 2013
  • Mathematica
    Select[Range[0, 7000], IntegerQ[Sqrt[13 # + 1]] &]
CoefficientList[Series[x (11 + 4 x + 11 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)

Formula

G.f.: x^2*(11+4*x+11*x^2)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = (26*n*(n-1)+9*(-1)^n*(2*n-1)+1)/8 +1.
Sum_{n>=2} 1/a(n) = 13/4 - cot(2*Pi/13)*Pi/2. - Amiram Eldar, Mar 15 2022

A219390 Numbers k such that 14*k+1 is a square.

Original entry on oeis.org

0, 12, 16, 52, 60, 120, 132, 216, 232, 340, 360, 492, 516, 672, 700, 880, 912, 1116, 1152, 1380, 1420, 1672, 1716, 1992, 2040, 2340, 2392, 2716, 2772, 3120, 3180, 3552, 3616, 4012, 4080, 4500, 4572, 5016, 5092, 5560, 5640, 6132, 6216, 6732, 6820
Offset: 1

Views

Author

Bruno Berselli, Nov 19 2012

Keywords

Comments

Equivalently, numbers of the form m*(14*m+2), where m = 0,-1,1,-2,2,-3,3,...
Also, integer values of 2*h*(h+1)/7.

Links

Crossrefs

Cf. similar sequences listed in A219257.
Cf. A113801 (square roots of 14*a(n)+1).

Programs

  • Magma
    [n: n in [0..7000] | IsSquare(14*n+1)];
  • Magma
    I:=[0,12,16,52,60]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013
  • Maple
    A219390:=proc(q)
local n;
for n from 1 to q do if type(sqrt(14*n+1), integer) then print(n);
fi; od; end:
A219390(1000); # Paolo P. Lava, Feb 19 2013
  • Mathematica
    Select[Range[0, 7000], IntegerQ[Sqrt[14 # + 1]] &]
CoefficientList[Series[4 x (3 + x + 3 x^2) ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,12,16,52,60},50] (* Harvey P. Dale, Feb 05 2019 *)

Formula

G.f.: 4*x^2*(3+x+3*x^2)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = (14*n*(n-1)+5*(-1)^n*(2*n-1)+1)/4 +1.
a(n) = 2*A219191(n).
Sum_{n>=2} 1/a(n) = 7/2 - cot(Pi/7)*Pi/2. - Amiram Eldar, Mar 15 2022

A219392 Numbers k such that 22*k+1 is a square.

Original entry on oeis.org

0, 20, 24, 84, 92, 192, 204, 344, 360, 540, 560, 780, 804, 1064, 1092, 1392, 1424, 1764, 1800, 2180, 2220, 2640, 2684, 3144, 3192, 3692, 3744, 4284, 4340, 4920, 4980, 5600, 5664, 6324, 6392, 7092, 7164, 7904, 7980, 8760, 8840, 9660, 9744, 10604, 10692
Offset: 1

Views

Author

Bruno Berselli, Nov 22 2012

Keywords

Comments

Equivalently, numbers of the form m*(22*m+2), where m = 0,-1,1,-2,2,-3,3,...
Also, integer values of 2*h*(h+1)/11.

Links

Crossrefs

Cf. similar sequences listed in A219257.

Programs

  • Magma
    [n: n in [0..11000] | IsSquare(22*n+1)];
  • Magma
    I:=[0,20,24,84,92]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013
  • Maple
    A219392:=proc(q)
local n;
for n from 1 to q do if type(sqrt(22*n+1), integer) then print(n);
fi; od; end:
A219392(1000); # Paolo P. Lava, Feb 19 2013
  • Mathematica
    Select[Range[0, 11000], IntegerQ[Sqrt[22 # + 1]] &]
CoefficientList[Series[4 x (5 + x + 5 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)

Formula

G.f.: 4*x^2*(5 + x + 5*x^2)/((1 + x)^2*(1 - x)^3).
a(n) = a(-n+1) = (22*n*(n-1) + 9*(-1)^n*(2*n - 1) + 1)/4 + 2.
Sum_{n>=2} 1/a(n) = 11/2 - cot(Pi/11)*Pi/2. - Amiram Eldar, Mar 16 2022

A219393 Numbers k such that 23*k+1 is a square.

Original entry on oeis.org

0, 21, 25, 88, 96, 201, 213, 360, 376, 565, 585, 816, 840, 1113, 1141, 1456, 1488, 1845, 1881, 2280, 2320, 2761, 2805, 3288, 3336, 3861, 3913, 4480, 4536, 5145, 5205, 5856, 5920, 6613, 6681, 7416, 7488, 8265, 8341, 9160, 9240, 10101, 10185, 11088, 11176, 12121, 12213
Offset: 1

Views

Author

Bruno Berselli, Nov 24 2012

Keywords

Comments

Equivalently, numbers of the form m*(23*m+2), where m = 0,-1,1,-2,2,-3,3,...
Also, integer values of h*(h+2)/23.

Links

Crossrefs

Cf. similar sequences listed in A219257.

Programs

  • Magma
    [n: n in [0..13000] | IsSquare(23*n+1)];
  • Magma
    I:=[0,21,25,88,96]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013
  • Maple
    A219393:=proc(q)
local n;
for n from 1 to q do if type(sqrt(23*n+1), integer) then print(n);
fi; od; end:
A219393(1000); # Paolo P. Lava, Feb 19 2013
  • Mathematica
    Select[Range[0, 13000], IntegerQ[Sqrt[23 # + 1]] &]
CoefficientList[Series[x (21 + 4 x + 21 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,21,25,88,96},50] (* Harvey P. Dale, Jun 22 2025 *)

Formula

G.f.: x^2*(21 + 4*x + 21*x^2)/((1 + x)^2*(1 - x)^3).
a(n) = a(-n+1) = (46*n*(n-1) + 19*(-1)^n*(2*n - 1) + 3)/8 + 2.
Sum_{n>=2} 1/a(n) = 23/4 - cot(2*Pi/23)*Pi/2. - Amiram Eldar, Mar 16 2022

A219394 Numbers k such that 17*k+1 is a square.

Original entry on oeis.org

0, 15, 19, 64, 72, 147, 159, 264, 280, 415, 435, 600, 624, 819, 847, 1072, 1104, 1359, 1395, 1680, 1720, 2035, 2079, 2424, 2472, 2847, 2899, 3304, 3360, 3795, 3855, 4320, 4384, 4879, 4947, 5472, 5544, 6099, 6175, 6760, 6840, 7455, 7539, 8184, 8272, 8947
Offset: 1

Views

Author

Bruno Berselli, Dec 03 2012

Keywords

Comments

Equivalently, numbers of the form m*(17*m+2), where m = 0,-1,1,-2,2,-3,3,...
Also, integer values of h*(h+2)/17.

Links

Crossrefs

Cf. similar sequences listed in A219257.

Programs

  • Magma
    [n: n in [0..9000] | IsSquare(17*n+1)];
  • Magma
    I:=[0,15,19,64,72]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013
  • Maple
    A219394:=proc(q)
local n;
for n from 1 to q do if type(sqrt(17*n+1), integer) then print(n);
fi; od; end:
A219394(1000); # Paolo P. Lava, Feb 19 2013
  • Mathematica
    Select[Range[0, 9000], IntegerQ[Sqrt[17 # + 1]] &]
CoefficientList[Series[x (15 + 4 x + 15 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,15,19,64,72},50] (* Harvey P. Dale, May 01 2017 *)

Formula

G.f.: x^2*(15+4*x+15*x^2)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = (34*n*(n-1)+13*(-1)^n*(2*n-1)+5)/8 + 1.
Sum_{n>=2} 1/a(n) = 17/4 - cot(2*Pi/17)*Pi/2. - Amiram Eldar, Mar 15 2022

A219395 Numbers k such that 18*k+1 is a square.

Original entry on oeis.org

0, 16, 20, 68, 76, 156, 168, 280, 296, 440, 460, 636, 660, 868, 896, 1136, 1168, 1440, 1476, 1780, 1820, 2156, 2200, 2568, 2616, 3016, 3068, 3500, 3556, 4020, 4080, 4576, 4640, 5168, 5236, 5796, 5868, 6460, 6536, 7160, 7240, 7896, 7980, 8668, 8756, 9476, 9568
Offset: 1

Views

Author

Bruno Berselli, Dec 03 2012

Keywords

Comments

Equivalently, numbers of the form m*(18*m+2), where m = 0,-1,1,-2,2,-3,3,...
Also, integer values of 2*h*(h+1)/9.
The sequence terms are the exponents in the expansion of Product_{n >= 1} (1 - q^(36*n))*(1 - q^(36*n-16))*(1 - q^(36*n-20)) = 1 - q^16 - q^20 + q^68 + q^76 - q^156 - q^168 + + - - .... - Peter Bala, Dec 24 2024

Links

Crossrefs

Cf. similar sequences listed in A219257.

Programs

  • Magma
    [n: n in [0..10000] | IsSquare(18*n+1)];
  • Magma
    I:=[0,16,20,68,76]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013
  • Maple
    A219395:=proc(q)
local n;
for n from 1 to q do if type(sqrt(18*n+1), integer) then print(n);
fi; od; end:
A219395(1000); # Paolo P. Lava, Feb 19 2013
  • Mathematica
    Select[Range[0, 10000], IntegerQ[Sqrt[18 # + 1]] &]
CoefficientList[Series[4 x (4 + x + 4 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,16,20,68,76},50] (* Harvey P. Dale, Dec 24 2014 *)

Formula

G.f.: 4*x^2*(4 + x + 4*x^2)/((1 + x)^2*(1 - x)^3).
a(n) = a(-n+1) = (18*n*(n-1) + 7*(-1)^n*(2*n-1) - 1)/4 + 2.
Sum_{n>=2} 1/a(n) = 9/2 - cot(Pi/9)*Pi/2. - Amiram Eldar, Mar 15 2022

A219396 Numbers k such that 19*k+1 is a square.

Original entry on oeis.org

0, 17, 21, 72, 80, 165, 177, 296, 312, 465, 485, 672, 696, 917, 945, 1200, 1232, 1521, 1557, 1880, 1920, 2277, 2321, 2712, 2760, 3185, 3237, 3696, 3752, 4245, 4305, 4832, 4896, 5457, 5525, 6120, 6192, 6821, 6897, 7560, 7640, 8337, 8421, 9152, 9240
Offset: 1

Views

Author

Bruno Berselli, Dec 03 2012

Keywords

Comments

Equivalently, numbers of the form m*(19*m+2), where m = 0, -1, 1, -2, 2, -3, 3,...
Also, integer values of h*(h+2)/19.

Links

Crossrefs

Cf. similar sequences listed in A219257.

Programs

  • Magma
    [n: n in [0..10000] | IsSquare(19*n+1)];
  • Magma
    I:=[0, 17, 21, 72, 80]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013
  • Maple
    A219396:=proc(q)
local n;
for n from 1 to q do if type(sqrt(19*n+1), integer) then print(n);
fi; od; end:
A219396(1000); # Paolo P. Lava, Feb 19 2013
  • Mathematica
    Select[Range[0, 10000], IntegerQ[Sqrt[19 # + 1]] &]
CoefficientList[Series[x (17 + 4 x + 17 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,17,21,72,80},60] (* Harvey P. Dale, Sep 08 2021 *)

Formula

G.f.: x^2*(17 + 4*x + 17*x^2)/((1 + x)^2*(1 - x)^3).
a(n) = a(-n+1) = (38*n*(n-1) + 15*(-1)^n*(2*n - 1)-1)/8 + 2.
Sum_{n>=2} 1/a(n) = 19/4 - cot(2*Pi/19)*Pi/2. - Amiram Eldar, Mar 15 2022

Extensions

Typo corrected in the first comment by Mokhtar Mohamed, Dec 03 2012

A157716 One-eighth of triangular numbers (integers only).

Original entry on oeis.org

0, 15, 17, 62, 66, 141, 147, 252, 260, 395, 405, 570, 582, 777, 791, 1016, 1032, 1287, 1305, 1590, 1610, 1925, 1947, 2292, 2316, 2691, 2717, 3122, 3150, 3585, 3615, 4080, 4112, 4607, 4641, 5166, 5202, 5757, 5795, 6380, 6420, 7035, 7077, 7722, 7766, 8441
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Mar 04 2009

Keywords

Comments

From Lamine Ngom, Oct 27 2020: (Start)
Numbers of the form (4*k)^2-k (A157446) or (4*k)^2+k (A157474).
Also numbers k such that 1+64*k is a square. (End)
The sequence terms are the exponents in the expansion of Product_{n >= 1} (1 - q^(32*n))*(1 - q^(32*n-15))*(1 - q^(32*n-17)) = 1 - q^15 - q^17 + q^62 + q^66 - q^141 - q^147 + + - - .... - Peter Bala, Dec 24 2024

Examples

			The first three members of A000217 that are divisible by 8 are A000217(0), A000217(15) and A000217(16), so a(1) = A000217(0)/8 = 0, a(2) = A000217(15)/8 = 15, a(3) = A000217(16)/8 = 17.

Links

Crossrefs

Cf. A000217, A101877, A219257.

Programs

  • Maple
    seq((2*n-1 + 7/8*(-1)^n)^2 - 1/64, n = 1 .. 1000); # Robert Israel, Apr 20 2014
  • Mathematica
    Array[(2 # - 1 + 7/8*(-1)^#)^2 - 1/64 &, 46] (* or *)
Rest@ CoefficientList[Series[x^2*(15 + 2 x + 15 x^2)/((1 + x)^2*(1 - x)^3), {x, 0, 46}], x] (* Michael De Vlieger, Nov 05 2020 *)

Formula

G.f.: x^2*(15+2*x+15*x^2)/((1+x)^2*(1-x)^3 ). [Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009; checked and corrected by R. J. Mathar, Sep 16 2009]
a(n) = (2*n-1 + 7/8*(-1)^n)^2 -1/64. - Robert Israel, Apr 20 2014
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Wesley Ivan Hurt, Nov 10 2020
Sum_{n>=2} 1/a(n) = 16 - (sqrt(2*(2+sqrt(2))) + sqrt(2) + 1)*Pi. - Amiram Eldar, Mar 17 2022

Extensions

Definition edited by N. J. A. Sloane, Mar 08 2009
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