A140676 -id:A140676 - cp's OEIS frontend

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

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)

A270710 a(n) = 3n^2 + 2n - 1.

Original entry on oeis.org

-1, 4, 15, 32, 55, 84, 119, 160, 207, 260, 319, 384, 455, 532, 615, 704, 799, 900, 1007, 1120, 1239, 1364, 1495, 1632, 1775, 1924, 2079, 2240, 2407, 2580, 2759, 2944, 3135, 3332, 3535, 3744, 3959, 4180, 4407, 4640, 4879, 5124, 5375, 5632, 5895, 6164, 6439, 6720, 7007, 7300, 7599

Offset: 0

Ilya Gutkovskiy, Mar 22 2016

In general, the ordinary generating function for the values of quadratic polynomial p*n^2 + q*n + k, is (k + (p + q - 2*k)*x + (p - q + k)*x^2)/(1 - x)^3.
From Bruno Berselli, Mar 25 2016: (Start)
This sequence and A140676 provide all integer m such that 3*m + 4 is a square.
Numbers related to A135713 by A135713(n) = n*a(n) - Sum_{k=0..n-1} a(k).
After -1, second bisection of A184005. (End)

			a(0) = 3*0^2 + 2*0 - 1 = -1;
a(1) = 3*1^2 + 2*1 - 1 =  4;
a(2) = 3*2^2 + 2*2 - 1 = 15;
a(3) = 3*3^2 + 2*3 - 1 = 32, etc.

Ilya Gutkovskiy, Examples of the ordinary generating function for the values of quadratic polynomial.
Leo Tavares, Triple Diamond Illustration.
Eric Weisstein's World of Mathematics, Quadratic Polynomial.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A033428, A045944, A056105, A056109, A060747, A135713, A140676, A184005.

Cf. A000290.

List([0..50], n -> 3*n^2+2*n-1); # Bruno Berselli, Feb 16 2018

[3*n^2+2*n-1: n in [0..50]]; // Bruno Berselli, Mar 25 2016

Table[3 n^2 + 2 n - 1, {n, 0, 50}]
LinearRecurrence[{3, -3, 1}, {-1, 4, 15}, 51]

makelist(3*n^2+2*n-1, n, 0, 50); /* Bruno Berselli, Mar 25 2016 */

Vec((-1 + 7*x)/(1 - x)^3 + O(x^60)) \\ Michel Marcus, Mar 22 2016

lista(nn) = {for(n=0, nn, print1(3*n^2 + 2*n - 1, ", ")); } \\ Altug Alkan, Mar 25 2016

vector(50, n, n--; 3*n^2+2*n-1) \\ Bruno Berselli, Mar 25 2016

[3*n^2+2*n-1 for n in (0..50)] # Bruno Berselli, Mar 25 2016

G.f.: (-1 + 7*x)/(1 - x)^3.
E.g.f.: exp(x)*(-1 + 5*x + 3*x^2).
a(n)  = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n)  = A033428(n) + A060747(n).
a(n)  = A045944(n) - 1 = A056109(n) - 2.
a(-n) = A140676(n-1), with A140676(-1) = -1.
Sum_{n>=0} 1/a(n) = 3*(log(3) - 2)/8 - Pi/(8*sqrt(3)) = -0.564745312278736...
a(n) = Sum_{i = n-1..2*n-1} (2*i + 1). - Bruno Berselli, Feb 16 2018
a(n) = A000290(n+1) + 2*A000290(n) - 2. - Leo Tavares, May 28 2023
Sum_{n>=0} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)) + 3/4. - Amiram Eldar, Jul 20 2023

A243201 Odd octagonal numbers indexed by triangular numbers.

Original entry on oeis.org

1, 21, 133, 481, 1281, 2821, 5461, 9633, 15841, 24661, 36741, 52801, 73633, 100101, 133141, 173761, 223041, 282133, 352261, 434721, 530881, 642181, 770133, 916321, 1082401, 1270101, 1481221, 1717633, 1981281, 2274181, 2598421, 2956161, 3349633, 3781141, 4253061, 4767841, 5328001

Offset: 0

Mathew Englander, Jun 01 2014

			a(2) = 133 because the second triangular number is 3 and third odd octagonal number is 133.
a(3) = 481 because the third triangular number is 6 and the sixth odd octagonal number is 481.
a(4) = 1281 because the fourth triangular number is 10 and the tenth odd octagonal number is 1281.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Row 5 of A059259 (coefficients of 1 + 4*n + 7*n^2 + 6*n^3 + 3*n^4 + 0*n^5 which is a formula for the within sequence).
Column 5 of A081297.
Column 6 of A072024.
Diagonal T(n + 1, n) of A219069, n > 0.
Cf. A000217, A000567, A002061, A003215, A005408, A014641, A022522, A140676, A187156.

[3*n^4+6*n^3+7*n^2+4*n+1: n in [0..40]]; // Bruno Berselli, Jun 03 2014

Table[((3 n^2 + 3 n + 2)^2 - 1)/3, {n, 0, 39}] (* Alonso del Arte, Jun 01 2014 *)

[3*n^4+6*n^3+7*n^2+4*n+1 for n in (0..40)] # Bruno Berselli, Jun 03 2014

a(n) = 3*n^4 + 6*n^3 + 7*n^2 + 4*n + 1.
a(n) = (n^2 + n + 1)*(3*n^2 + 3*n + 1).
a(n) = ((3*n^2 + 3*n + 2)^2 - 1)/3.
a(n) = A003215(n) * A002061(n + 1).
a(n) = A022522(n) / A005408(n).
a(n) = A000567(n^2 + n + 1).
a(n) = A014641((n^2 + n)/2).
a(n) = 1 + A140676(n^2 + n).
a(n) = 1 + A187156((n^2 + n + 4)/2) (empirical).
G.f.: (1 + 16*x + 38*x^2 + 16*x^3 + x^4)/(1 - x)^5. - Bruno Berselli, Jun 03 2014
E.g.f.: exp(x)*(1 + 20*x + 46*x^2 + 24*x^3 + 3*x^4). - Stefano Spezia, Apr 16 2022

A271675 Numbers m such that 3*m + 4 is a square.

Original entry on oeis.org

0, 4, 7, 15, 20, 32, 39, 55, 64, 84, 95, 119, 132, 160, 175, 207, 224, 260, 279, 319, 340, 384, 407, 455, 480, 532, 559, 615, 644, 704, 735, 799, 832, 900, 935, 1007, 1044, 1120, 1159, 1239, 1280, 1364, 1407, 1495, 1540, 1632, 1679, 1775, 1824, 1924, 1975, 2079, 2132, 2240, 2295, 2407

Offset: 1

Juri-Stepan Gerasimov, Apr 12 2016

7 is the unique prime in this sequence. If m is in this sequence, then 3*m + 4 = k^2 for k is nonzero integer, that is, m = (k^2 - 4)/3 = (k-2)*(k+2)/3. So m can be only prime if one of divisors is prime and another one is 1. Otherwise there should be more than 1 prime divisors, that is n must be composite. - Altug Alkan, Apr 12 2016
From Ray Chandler, Apr 12 2016: (Start)
Square roots of resulting squares gives A001651 (with a different starting point).
Sequence is the union of (positive terms) in A140676 and A270710. (End)
The sequence terms are the exponents in the expansion of Sum_{n >= 0} q^n*(1 - q)*(1 - q^3)*...*(1 - q^(2*n+1)) = 1 - q^4 - q^7 + q^15 + q^20 - q^32 - q^50 + + - - .... - Peter Bala, Dec 19 2024

			a(4) = 32 because 3*32 + 4 = 100 = 10*10.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A001651, A140676, A270710.

Cf. numbers n such that 3*n + k is a square: A120328 (k=-6), A271713 (k=-5), A056107 (k=-3), A257083 (k=-2), A033428 (k=0), A001082 (k=1), A080663 (k=3), this sequence (k=4), A100536 (k=6).

Magma
```
[n: n in [0..4000] | IsSquare(3*n+4)];
```

Select[Range[0,2500], IntegerQ@ Sqrt[3 # + 4] &] (* Michael De Vlieger, Apr 12 2016 *)
LinearRecurrence[{1,2,-2,-1,1},{0,4,7,15,20},60] (* Harvey P. Dale, Dec 09 2016 *)

from gmpy2 import is_square
for n in range(0,10**5):
    if(is_square(3*n+4)):print(n)
# Soumil Mandal, Apr 12 2016

O.g.f.: x^2*(4 + 3*x - x^3)/((1 + x)^2*(1 - x)^3).
E.g.f.: 1 + (1 - 2*x)*exp(-x)/8 - 3*(3 - 4*x - 2*x^2)*exp(x)/8.
a(n) = A001082(n+1) - 1 = (6*n*(n+1) + (2*n + 1)*(-1)^n - 1)/8 - 1. Therefore: a(2*k+1) = k*(3*k+4), a(2*k) = (k+1)*(3*k-1).
Sum_{n>=2} 1/a(n) = 19/16 - Pi/(4*sqrt(3)). - Amiram Eldar, Jul 26 2024

Edited and extended by Bruno Berselli, Apr 12 2016

A182866 Number of edges in the n^2 X n^2 Sudoku graph.

Original entry on oeis.org

0, 56, 810, 4992, 20000, 61560, 158466, 358400, 734832, 1395000, 2488970, 4219776, 6854640, 10737272, 16301250, 24084480, 34744736, 49076280, 68027562, 92720000, 124467840, 164799096, 215477570, 278525952, 356250000, 451263800, 566516106, 705317760

Offset: 1

Douglas Smith, Feb 01 2011

			For the standard Sudoku (n=3) there are 81 vertices. Each vertex is connected to 8 others within its own square, and 12 others in its row and column. Dividing by 2 gives 810 edges.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000583 (number of vertices), A140676 (degree of each vertex).

Table[n^4/2 (-1-2n+3n^2),{n,30}] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{0,56,810,4992,20000,61560,158466},30] (* Harvey P. Dale, Oct 30 2011 *)

Vec(2*x^2*(28+209*x+249*x^2+53*x^3+x^4)/(1-x)^7+O(x^99)) \\ Charles R Greathouse IV, Jan 25 2012

a(n) = (1/2)*n^4*(-1 - 2*n + 3*n^2).
a(1)=0, a(2)=56, a(3)=810, a(4)=4992, a(5)=20000, a(6)=61560, a(7)=158466, a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). - Harvey P. Dale, Oct 30 2011
G.f.: 2*x^2*(28 + 209*x + 249*x^2 + 53*x^3 + x^4)/(1-x)^7. - Colin Barker, Jan 25 2012

cp's OEIS Frontend

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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A270710 a(n) = 3*n^2 + 2*n - 1.

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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).

A270710 a(n) = 3n^2 + 2n - 1.