A067727 -id:A067727 - cp's OEIS frontend

A067725 a(n) = 3n^2 + 6n.

0, 9, 24, 45, 72, 105, 144, 189, 240, 297, 360, 429, 504, 585, 672, 765, 864, 969, 1080, 1197, 1320, 1449, 1584, 1725, 1872, 2025, 2184, 2349, 2520, 2697, 2880, 3069, 3264, 3465, 3672, 3885, 4104, 4329, 4560, 4797, 5040, 5289, 5544, 5805, 6072, 6345, 6624

Offset: 0

Robert G. Wilson v, Feb 05 2002

Numbers h such that 3*(3 + h) is a perfect square. - Alexander D. Healy, Tj Tullo, Avery Pickford, Sep 20 2004

Equivalently, numbers k such that k/3+1 is a square. - Bruno Berselli, Apr 10 2018

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005563.

Cf. numbers k such that k*(k + m) is a perfect square: A028560 (k=9), A067728 (k=8), A067727 (k=7), A067726 (k=6), A067724 (k=5), A028347 (k=4), A054000 (k=2), A005563 (k=1).

List([0..50], n-> 3*n*(n+2)); # G. C. Greubel, Sep 01 2019

[3*n*(n+2): n in [0..50]]; // G. C. Greubel, Sep 01 2019

seq(3*n*(n+2), n=0..50); # G. C. Greubel, Sep 01 2019

Select[ Range[10000], IntegerQ[ Sqrt[ 3(3 + # )]] & ]
3*(Range[50]^2 -1) (* G. C. Greubel, Sep 01 2019 *)

a(n)=3*n*(n+2) \\ Charles R Greathouse IV, Dec 07 2011

[3*n*(n+2) for n in (0..50)] # G. C. Greubel, Sep 01 2019

a(n) = 3*A005563(n). - Zerinvary Lajos, Mar 06 2007
a(n) = a(n-1) + 6*n + 3, with n>0, a(0)=0. - Vincenzo Librandi, Aug 08 2010
From Colin Barker, Apr 11 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: 3*x*(3-x)/(1-x)^3. (End)
E.g.f.: 3*x*(x + 3)*exp(x). - G. C. Greubel, Jul 20 2017
From Amiram Eldar, Feb 26 2022: (Start)
Sum_{n>=1} 1/a(n) = 1/4.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/12. (End)

Edited by N. J. A. Sloane, Sep 14 2008 at the suggestion of R. J. Mathar

A067728 a(n) = 2n^2 + 8n.

Original entry on oeis.org

10, 24, 42, 64, 90, 120, 154, 192, 234, 280, 330, 384, 442, 504, 570, 640, 714, 792, 874, 960, 1050, 1144, 1242, 1344, 1450, 1560, 1674, 1792, 1914, 2040, 2170, 2304, 2442, 2584, 2730, 2880, 3034, 3192, 3354, 3520, 3690, 3864, 4042, 4224, 4410, 4600, 4794

Offset: 1

Robert G. Wilson v, Feb 05 2002

Positive numbers k such that 8*(8 + k) is a perfect 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. 7: A067727, 6: A067726, 5: A067724, 3: A067725.

Cf. A000217, A005563, A140091, A140681, A212331.

[2*n*(n+4): n in [1..50]] // Vincenzo Librandi, Jul 08 2012

Select[ Range[10000], IntegerQ[ Sqrt[ 8(8 + # )]] & ]
CoefficientList[Series[2*(5-3*x)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 08 2012 *)

a(n)=2*n*(n+4) \\ Charles R Greathouse IV, Dec 07 2011

def a(n): return (2*n + 8)*n
print([a(n) for n in range(1, 48)]) # Michael S. Branicky, Oct 24 2021

a(n+1) = 2*n*n + 12*n + 10. - Frank Ellermann
a(n) = Sum_{k=0..n} Sum_{j=4..n} (j - k), n >= 4. - Zerinvary Lajos, May 11 2007
From Vincenzo Librandi, Jul 08 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: 2*x*(5-3*x)/(1-x)^3. (End)
From Amiram Eldar, Feb 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 25/96.
Sum_{n>=1} (-1)^(n+1)/a(n) = 7/96. (End)
E.g.f.: 2*exp(x)*x*(5 + x). - Stefano Spezia, Oct 01 2023

A186029 a(n) = n(7n+3)/2.

Original entry on oeis.org

0, 5, 17, 36, 62, 95, 135, 182, 236, 297, 365, 440, 522, 611, 707, 810, 920, 1037, 1161, 1292, 1430, 1575, 1727, 1886, 2052, 2225, 2405, 2592, 2786, 2987, 3195, 3410, 3632, 3861, 4097, 4340, 4590, 4847, 5111, 5382, 5660, 5945, 6237, 6536, 6842, 7155, 7475

Offset: 0

Bruno Berselli, Feb 11 2011

This sequence is related to A050409 by A050409(n) = n*a(n) - Sum_{i=0..n-1} a(i).

			From _Ilya Gutkovskiy_, Mar 31 2016: (Start)
.                                           o o o o o o o o o o o o
.                                           o                     o
.         o o o o o o   o  o o o o o o  o   o  o  o o o o o o  o  o
.         o         o   o  o         o  o   o  o  o         o  o  o
. o   o   o  o   o  o   o  o  o   o  o  o   o  o  o  o   o  o  o  o
. o o o   o  o o o  o   o  o  o o o  o  o   o  o  o  o o o  o  o  o
.                       o               o   o  o               o  o
.                       o o o o o o o o o   o  o o o o o o o o o  o
.
.  n=1        n=2              n=3                    n=4
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. numbers of the form n*(d*n+10-d)/2 indexed in A140090.
Cf. A017041 (first differences).
Cf. A001106, A022264, A022265, A024966, A050409, A067727, A174738, A179986, A218471.

Magma
```
[n*(7*n+3)/2: n in [0..44]];
```

Table[(n - 1) (7 n - 4)/2, {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2011 *)
LinearRecurrence[{3,-3,1},{0,5,17},50] (* Harvey P. Dale, Sep 07 2022 *)

a(n)=n*(7*n+3)/2 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: x*(5+2*x)/(1-x)^3.
a(n) - a(-n) = A008585(n).
a(n) + a(-n) = A033582(n).
n*a(n+1) - (n+1)*a(n) = A024966(n). - Bruno Berselli, May 30 2012
n*a(n+2) - (n+2)*a(n) = A067727(n) for n>0. - Bruno Berselli, May 30 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0)=0, a(1)=5, a(2)=17. - Philippe Deléham, Mar 26 2013
a(n) = A174738(7*n+4). - Philippe Deléham, Mar 26 2013
E.g.f.: (1/2)*(7*x^2 + 10*x)*exp(x). - G. C. Greubel, Jul 17 2017

A067707 a(n) = 3n^2 + 12n.

Original entry on oeis.org

15, 36, 63, 96, 135, 180, 231, 288, 351, 420, 495, 576, 663, 756, 855, 960, 1071, 1188, 1311, 1440, 1575, 1716, 1863, 2016, 2175, 2340, 2511, 2688, 2871, 3060, 3255, 3456, 3663, 3876, 4095, 4320, 4551, 4788, 5031, 5280, 5535, 5796, 6063, 6336, 6615, 6900

Offset: 1

Robert G. Wilson v, Feb 05 2002

Numbers k such that 12*(12 + k) is a perfect square.

a(n) is the second Zagreb index of the gear graph g[n]. The second Zagreb index of a simple connected graph is the sum of the degree products d(i)d(j) over all edges ij of the graph. The gear graph g[n] is defined as a wheel graph with n+1 vertices with a vertex added between each pair of adjacent vertices of the outer cycle. - Emeric Deutsch, Nov 09 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Gear Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A067724 (5), A067725 (3), A067726 (6), A067727 (7), A067728, A067705 (11).

[3*n^2 + 12*n: n in [1..50]]; // Vincenzo Librandi, Jul 07 2012

Select[ Range[10000], IntegerQ[ Sqrt[ 12(12 + # )]] & ]
CoefficientList[Series[3*(5-3*x)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 07 2012 *)

a(n)=3*n*(n+4) \\ Charles R Greathouse IV, Dec 07 2011

G.f.: 3*x*(5 - 3*x)/(1 - x)^3. - Vincenzo Librandi, Jul 07 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 07 2012
E.g.f.: 3*x*(x + 5)*exp(x). - G. C. Greubel, Jul 20 2017
From Amiram Eldar, Feb 26 2022: (Start)
Sum_{n>=1} 1/a(n) = 25/144.
Sum_{n>=1} (-1)^(n+1)/a(n) = 7/144. (End)

A067724 a(n) = 5n^2 + 10n.

Original entry on oeis.org

15, 40, 75, 120, 175, 240, 315, 400, 495, 600, 715, 840, 975, 1120, 1275, 1440, 1615, 1800, 1995, 2200, 2415, 2640, 2875, 3120, 3375, 3640, 3915, 4200, 4495, 4800, 5115, 5440, 5775, 6120, 6475, 6840, 7215, 7600, 7995, 8400, 8815, 9240, 9675

Offset: 1

Robert G. Wilson v, Feb 05 2002

Positive numbers m such that 5*(5 + m) is a perfect 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. numbers k such that k*(k + m) is a perfect square: A028560 (k=9), A067728 (k=8), A067727 (k=7), A067726 (k=6), A028347 (k=4), A067725 (k=3), A054000 (k=2), A067998 (k=1).

Cf. A055998.

[5*n*(n+2): n in [1..50]]; // Vincenzo Librandi, Jul 08 2012

Select[Range[10000], IntegerQ[ Sqrt[5 (5 + # )]] &]
CoefficientList[Series[5 (3 - x)/(1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 08 2012 *)
Table[5n^2+10n,{n,60}] (* or *) LinearRecurrence[{3,-3,1},{15,40,75},60] (* Harvey P. Dale, May 22 2018 *)

a(n)=5*n*(n+2) \\ Charles R Greathouse IV, Dec 07 2011

From Vincenzo Librandi, Jul 08 2012: (Start)
G.f.: 5*x*(3 - x)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
a(n) = A055998(3*n) + A055998(n). - Bruno Berselli, Sep 23 2016
From Amiram Eldar, Feb 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 3/20.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/20. (End)
E.g.f.: 5*exp(x)*x*(3 + x). - Stefano Spezia, Oct 01 2023

A067726 a(n) = 6n^2 + 12n.

Original entry on oeis.org

18, 48, 90, 144, 210, 288, 378, 480, 594, 720, 858, 1008, 1170, 1344, 1530, 1728, 1938, 2160, 2394, 2640, 2898, 3168, 3450, 3744, 4050, 4368, 4698, 5040, 5394, 5760, 6138, 6528, 6930, 7344, 7770, 8208, 8658, 9120, 9594, 10080, 10578, 11088, 11610

Offset: 1

Robert G. Wilson v, Feb 05 2002

Positive numbers k such that 6*(6 + k) is a perfect square.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Leo Tavares, Illustration: Star Cut Hexagons
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. numbers k such that k*(k + m) is a perfect square: A028560 (k=9), A067728 (k=8), A067727 (k=7), A067724 (k=5), A028347 (k=4), A067725 (k=3), A054000 (k=2), A005563 (k=1).

Cf. A003154, A003215.

List([1..45], n-> 6*n*(n+2)); # G. C. Greubel, Sep 01 2019

[6*n*(n+2): n in [1..50]]; // Vincenzo Librandi, Jul 08 2012

seq(6*n*(n+2), n=1..45); # G. C. Greubel, Sep 01 2019

Select[ Range[15000], IntegerQ[ Sqrt[ 6(6 + # )]] & ]
CoefficientList[Series[6*(3-x)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 08 2012 *)
6*(Range[2, 45]^2 -1) (* G. C. Greubel, Sep 01 2019 *)
LinearRecurrence[{3,-3,1},{18,48,90},60] (* Harvey P. Dale, May 10 2022 *)

a(n)=6*n*(n+2) \\ Charles R Greathouse IV, Dec 07 2011

[6*n*(n+2) for n in (1..45)] # G. C. Greubel, Sep 01 2019

G.f.: 6*x*(3 - x)/(1 - x)^3. - Vincenzo Librandi, Jul 08 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 08 2012
E.g.f.: 6*x*(3 + x)*exp(x). - G. C. Greubel, Sep 01 2019
From Amiram Eldar, Feb 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 1/8.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/24. (End)
a(n) = A003215(2*n) - A003154(n). - Leo Tavares, May 20 2023
a(n) = 6*A005563(n). - Hugo Pfoertner, May 24 2023

A067705 a(n) = 11n^2 + 22n.

Original entry on oeis.org

33, 88, 165, 264, 385, 528, 693, 880, 1089, 1320, 1573, 1848, 2145, 2464, 2805, 3168, 3553, 3960, 4389, 4840, 5313, 5808, 6325, 6864, 7425, 8008, 8613, 9240, 9889, 10560, 11253, 11968, 12705, 13464, 14245, 15048, 15873, 16720, 17589, 18480, 19393, 20328, 21285

Offset: 1

Robert G. Wilson v, Feb 05 2002

Numbers k such that 11*(11 + k) is a perfect 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. A067724, A067725, A067726, A067727, A067728 (if 11 is replaced by 3, 5, 6, 7, 8 respectively), A067707 (12).

Cf. A005563.

[11*n*(n+2): n in [1..50]]; // Vincenzo Librandi, Jul 07 2012

Select[ Range[20000], IntegerQ[ Sqrt[ 11(11 + # )]] & ]
CoefficientList[Series[11 (3 - x)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Jul 07 2012 *)

a(n)=11*n*(n+2) \\ Charles R Greathouse IV, Dec 07 2011

From Vincenzo Librandi, Jul 07 2012: (Start)
G.f.: 11*x*(3-x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Elmo R. Oliveira, Jan 28 2025: (Start)
E.g.f.: 11*exp(x)*x*(3 + x).
a(n) = 11*A005563(n). (End)

A302576 Numbers k such that k/10 + 1 is a square.

Original entry on oeis.org

-10, 0, 30, 80, 150, 240, 350, 480, 630, 800, 990, 1200, 1430, 1680, 1950, 2240, 2550, 2880, 3230, 3600, 3990, 4400, 4830, 5280, 5750, 6240, 6750, 7280, 7830, 8400, 8990, 9600, 10230, 10880, 11550, 12240, 12950, 13680, 14430, 15200, 15990, 16800, 17630, 18480, 19350, 20240

Offset: 1

Bruno Berselli, Apr 10 2018

Equivalently, numbers k such that (k + 10)*10 is a square.

The positive terms belong to the fourth column of the array in A185781.

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

Cf. A000290, A008592, A033583, A185781.
After -10, subsequence of A174133 because a(n) = ((n-1)^2-1)*(3^2+1).
Similar lists of k for which k/j + 1 is a square: A067998 (j=1), A054000 (j=2), A067725 (j=3), A134582 (j=4), A067724 (j=5), A067726 (j=6), A067727 (j=7), second bisection of A067728 (j=8), A147651 (j=9), this sequence (j=10), A067705 (j=11), second bisection of A067707 (j=12).

GAP
```
List([1..50], n -> 10*n*(n-2));
```
Julia
```
[10*n*(n-2) for n in 1:50] |> println
```
Magma
```
[10*n*(n-2): n in [1..50]];
```
Mathematica
```
Table[10 n (n - 2), {n, 1, 50}]
```
Maxima
```
makelist(10*n*(n-2), n, 1, 50);
```
PARI
```
vector(50, n, nn; 10*n*(n-2))
```
Python
```
[10*n*(n-2) for n in range(1, 50)]
```
Sage
```
[10*n*(n-2) for n in (1..50)]
```

O.g.f.: -10*x*(1 - 3*x)/(1 - x)^3.
E.g.f.: -10*x*(1 - x)*exp(x).
a(n) = a(2-n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 10*n*(n - 2) = 10*A067998(n).
a(n) = A033583(n-1) - 10. - Altug Alkan, Apr 10 2018

cp's OEIS Frontend

A067725 a(n) = 3*n^2 + 6*n.

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A067728 a(n) = 2*n^2 + 8*n.

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A186029 a(n) = n*(7*n+3)/2.

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A067707 a(n) = 3*n^2 + 12*n.

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A067724 a(n) = 5*n^2 + 10*n.

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A067726 a(n) = 6*n^2 + 12*n.

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A067725 a(n) = 3n^2 + 6n.

A067728 a(n) = 2n^2 + 8n.

A186029 a(n) = n(7n+3)/2.

A067707 a(n) = 3n^2 + 12n.

A067724 a(n) = 5n^2 + 10n.

A067726 a(n) = 6n^2 + 12n.

A067705 a(n) = 11n^2 + 22n.