A067707 -id:A067707 - cp's OEIS frontend

A140676 a(n) = n(3n + 4).

0, 7, 20, 39, 64, 95, 132, 175, 224, 279, 340, 407, 480, 559, 644, 735, 832, 935, 1044, 1159, 1280, 1407, 1540, 1679, 1824, 1975, 2132, 2295, 2464, 2639, 2820, 3007, 3200, 3399, 3604, 3815, 4032, 4255, 4484, 4719, 4960, 5207, 5460, 5719, 5984, 6255, 6532, 6815

Offset: 0

Omar E. Pol, May 22 2008

The number of peers of a cell of an n^2 X n^2 sudoku is a(n-1). - Neven Sajko, Apr 20 2016

First differences are in A016921. - Wesley Ivan Hurt, Apr 21 2016

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. A000567, A016921, A033428, A045944, A067707, A067725, A140677, A140678, A140679, A140680, A140681, A140689.

[n*(3*n+4) : n in [0..80]]; // Wesley Ivan Hurt, Apr 21 2016

A140676:=n->n*(3*n+4): seq(A140676(n), n=0..100); # Wesley Ivan Hurt, Apr 21 2016

Table[n (3 n + 4), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 7, 20}, 50] (* Harvey P. Dale, May 04 2013 *)

a(n)=n*(3*n+4) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 3*n^2 + 4*n.
a(n) = 6*n + a(n-1) + 1 for n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
O.g.f.: x*(7 - x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - Harvey P. Dale, May 04 2013
E.g.f.: x*(7 + 3*x)*exp(x). - Ilya Gutkovskiy, Apr 20 2016
a(n) = A000567(n+1) - 1. - Neven Sajko, Apr 20 2016
From Amiram Eldar, Feb 26 2022: (Start)
Sum_{n>=1} 1/a(n) = 15/16 - Pi/(8*sqrt(3)) - 3*log(3)/8.
Sum_{n>=1} (-1)^(n+1)/a(n) = 9/16 - Pi/(4*sqrt(3)). (End)

A140681 a(n) = 3n(n+6).

Original entry on oeis.org

0, 21, 48, 81, 120, 165, 216, 273, 336, 405, 480, 561, 648, 741, 840, 945, 1056, 1173, 1296, 1425, 1560, 1701, 1848, 2001, 2160, 2325, 2496, 2673, 2856, 3045, 3240, 3441, 3648, 3861, 4080, 4305, 4536, 4773, 5016, 5265, 5520, 5781

Offset: 0

Omar E. Pol, May 22 2008

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. A028560, A033428, A045944, A140676, A067725, A140677, A140678, A067707, A140679, A140680, A140689, A000217, A005563, A067728, A140091, A212331.

A140681:=n->3*n*(n+6); seq(A140681(n), n=0..100); # Wesley Ivan Hurt, Dec 10 2013

Table[3n(n+6), {n,0,100}] (* Wesley Ivan Hurt, Dec 10 2013 *)
LinearRecurrence[{3,-3,1},{0,21,48},50] (* Harvey P. Dale, May 03 2023 *)

a(n)=3*n*(n+6) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A028560(n)*3 = 3*n^2 + 18*n = n*(3*n+18).
a(n) = 6*n + a(n-1) + 15 with n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
from G. C. Greubel, Jul 20 2017: (Start)
G.f.: 3*x*(7 - 5*x)/(1-x)^3.
E.g.f.: 3*x*(x+7)*exp(x). (End)
From Amiram Eldar, Feb 26 2022: (Start)
Sum_{n>=1} 1/a(n) = 49/360.
Sum_{n>=1} (-1)^(n+1)/a(n) = 37/1080. (End)

A140677 a(n) = n(3n + 8).

Original entry on oeis.org

0, 11, 28, 51, 80, 115, 156, 203, 256, 315, 380, 451, 528, 611, 700, 795, 896, 1003, 1116, 1235, 1360, 1491, 1628, 1771, 1920, 2075, 2236, 2403, 2576, 2755, 2940, 3131, 3328, 3531, 3740, 3955, 4176, 4403, 4636, 4875, 5120, 5371, 5628

Offset: 0

Omar E. Pol, May 22 2008

			a(1) = 6*1 + 0 + 5 = 11; a(2) = 6*2 + 11 + 5 = 28; a(3) = 6*3 + 28 + 5 = 51. - _Vincenzo Librandi_, Aug 03 2010

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. A033428, A045944, A140676, A067725, A140678, A067707, A140679, A140680, A140681, A140689.

Table[n(3n+8),{n,0,45}]  (* Harvey P. Dale, Feb 20 2011 *)

a(n)=n*(3*n+8) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 3*n^2 + 8*n.
a(n) = 6*n + a(n-1) + 5, with a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: x*(11 - 5*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
E.g.f.: (3*x^2 + 11*x)*exp(x). - G. C. Greubel, Jul 20 2017

A140678 a(n) = n(3n + 10).

Original entry on oeis.org

0, 13, 32, 57, 88, 125, 168, 217, 272, 333, 400, 473, 552, 637, 728, 825, 928, 1037, 1152, 1273, 1400, 1533, 1672, 1817, 1968, 2125, 2288, 2457, 2632, 2813, 3000, 3193, 3392, 3597, 3808, 4025, 4248, 4477, 4712, 4953, 5200, 5453, 5712

Offset: 0

Omar E. Pol, May 22 2008

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

Cf. A033428, A045944, A140676, A067725, A140677, A067707, A140679, A140680, A140681, A140689.

Table[n (3 n + 10), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 13, 32}, 50] (* Harvey P. Dale, Jun 05 2012 *)

a(n)=n*(3*n+10) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 3*n^2 + 10*n.
a(n) = 6*n + a(n-1) + 7, with a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: x*(13 - 7*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(0)=0, a(1)=13, a(2)=32. - Harvey P. Dale, Jun 05 2012
E.g.f.: (3*x^2 + 13*x)*exp(x). - G. C. Greubel, Jul 20 2017

A140679 a(n) = n(3n+14).

Original entry on oeis.org

0, 17, 40, 69, 104, 145, 192, 245, 304, 369, 440, 517, 600, 689, 784, 885, 992, 1105, 1224, 1349, 1480, 1617, 1760, 1909, 2064, 2225, 2392, 2565, 2744, 2929, 3120, 3317, 3520, 3729, 3944, 4165, 4392, 4625, 4864, 5109, 5360, 5617, 5880, 6149, 6424, 6705, 6992

Offset: 0

Omar E. Pol, May 22 2008

			a(1)=6*1+0+11=17; a(2)=6*2+17+11=40; a(3)=6*3+40+11=69. See 2nd formula.

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. A033428, A045944, A140676, A067725, A140677, A140678, A067707, A140680, A140681, A140689.

Table[n(3n+14),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,17,40},50] (* Harvey P. Dale, Apr 29 2011 *)

a(n)=n*(3*n+14) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 3*n^2 + 14*n.
a(n) = a(n-1) + 6*n + 11, with a(0)=0. - Vincenzo Librandi, Aug 03 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(1)=0, a(2)=17, a(3)=40. - Harvey P. Dale, Apr 29 2011
E.g.f.: (3*x^2 + 17*x)*exp(x). - G. C. Greubel, Jul 20 2017

A140680 a(n) = n(3n+16).

Original entry on oeis.org

0, 19, 44, 75, 112, 155, 204, 259, 320, 387, 460, 539, 624, 715, 812, 915, 1024, 1139, 1260, 1387, 1520, 1659, 1804, 1955, 2112, 2275, 2444, 2619, 2800, 2987, 3180, 3379, 3584, 3795, 4012, 4235, 4464, 4699, 4940, 5187, 5440, 5699

Offset: 0

Omar E. Pol, May 22 2008

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. A033428, A045944, A140676, A067725, A140677, A140678, A067707, A140679, A140681, A140689.

a[n_]:=Sum[6*i+13, {i, 1, n}]; (* Vladimir Joseph Stephan Orlovsky, Dec 04 2008 *)
Table[n(3n+16),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,19,44},50] (* Harvey P. Dale, Aug 23 2020 *)

a(n)=n*(3*n+16) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 3*n^2 + 16*n.
a(n) = 6*n + a(n-1) + 13 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
E.g.f.: (3*x^2 + 19*x)*exp(x). - G. C. Greubel, Jul 20 2017

A140689 a(n) = n(3n + 20).

Original entry on oeis.org

0, 23, 52, 87, 128, 175, 228, 287, 352, 423, 500, 583, 672, 767, 868, 975, 1088, 1207, 1332, 1463, 1600, 1743, 1892, 2047, 2208, 2375, 2548, 2727, 2912, 3103, 3300, 3503, 3712, 3927, 4148, 4375, 4608, 4847, 5092, 5343, 5600, 5863

Offset: 0

Omar E. Pol, May 22 2008

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

Cf. A033428, A045944, A140676, A067725, A140677, A140678, A067707, A140679, A140680, A140681.

a[n_]:=Sum[6*i+17, {i, 1, n}]; (* Vladimir Joseph Stephan Orlovsky, Dec 04 2008 *)
Table[n(3n+20),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,23,52},50] (* Harvey P. Dale, Apr 29 2016 *)

a(n)=n*(3*n+20) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 3*n^2 + 20*n.
a(n) = a(n-1) + 6*n + 17 (with a(0)=0). - Vincenzo Librandi, Dec 15 2010
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3), with a(0)=0, a(1)=23, a(2)=52. - Harvey P. Dale, Apr 29 2016
From G. C. Greubel, Jul 21 2017: (Start)
G.f.: x*(23 - 17*x)/(1 - x)^3.
E.g.f.: x*(3*x + 23)*exp(x). (End)

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

A384125 Array read by antidiagonals: T(n,m) is the number of edges in the n X m rook graph K_n X K_m.

Original entry on oeis.org

0, 1, 1, 3, 4, 3, 6, 9, 9, 6, 10, 16, 18, 16, 10, 15, 25, 30, 30, 25, 15, 21, 36, 45, 48, 45, 36, 21, 28, 49, 63, 70, 70, 63, 49, 28, 36, 64, 84, 96, 100, 96, 84, 64, 36, 45, 81, 108, 126, 135, 135, 126, 108, 81, 45, 55, 100, 135, 160, 175, 180, 175, 160, 135, 100, 55

Offset: 1

Andrew Howroyd, May 20 2025

			Array begins:
=======================================
n\m |  1  2   3   4   5   6   7   8 ...
----+----------------------------------
  1 |  0  1   3   6  10  15  21  28 ...
  2 |  1  4   9  16  25  36  49  64 ...
  3 |  3  9  18  30  45  63  84 108 ...
  4 |  6 16  30  48  70  96 126 160 ...
  5 | 10 25  45  70 100 135 175 220 ...
  6 | 15 36  63  96 135 180 231 288 ...
  7 | 21 49  84 126 175 231 294 364 ...
  8 | 28 64 108 160 220 288 364 448 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
Eric Weisstein's World of Mathematics, Edge Count.
Eric Weisstein's World of Mathematics, Rook Graph.

Main diagonal is A045991.
Columns 1..6 are A000217(n-1), A000290, A045943, A054000, A269457(n-1), A067707.
Cf. A003991 (number of vertices), A360855 (triangles), A384120 (all cliques).

Table[#*Binomial[m, 2] + m*Binomial[#, 2] &[n - m + 1], {n, 11}, {m, n}] // Flatten (* Michael De Vlieger, May 22 2025 *)

T(n,m) = n*binomial(m,2) + m*binomial(n,2)

T(n,m) = n*binomial(m,2) + m*binomial(n,2).
T(n,m) = binomial(n*m,2) - 2*binomial(n,2)*binomial(m,2).
T(n,m) = T(m,n).

cp's OEIS Frontend

A140676 a(n) = n*(3*n + 4).

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A140681 a(n) = 3*n*(n+6).

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A140677 a(n) = n*(3*n + 8).

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A140678 a(n) = n*(3*n + 10).

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A140679 a(n) = n*(3*n+14).

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A140680 a(n) = n*(3*n+16).

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A140689 a(n) = n*(3*n + 20).

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

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A140676 a(n) = n(3n + 4).

A140681 a(n) = 3n(n+6).

A140677 a(n) = n(3n + 8).

A140678 a(n) = n(3n + 10).

A140679 a(n) = n(3n+14).

A140680 a(n) = n(3n+16).

A140689 a(n) = n(3n + 20).

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