A179986 -id:A179986 - cp's OEIS frontend

A135705 a(n) = 10binomial(n,2) + 9n.

0, 9, 28, 57, 96, 145, 204, 273, 352, 441, 540, 649, 768, 897, 1036, 1185, 1344, 1513, 1692, 1881, 2080, 2289, 2508, 2737, 2976, 3225, 3484, 3753, 4032, 4321, 4620, 4929, 5248, 5577, 5916, 6265, 6624, 6993, 7372, 7761, 8160, 8569, 8988, 9417, 9856, 10305, 10764

Offset: 0

N. J. A. Sloane, Mar 04 2008

Also, second 12-gonal (or dodecagonal) numbers. Identity for the numbers b(n)=n*(h*n+h-2)/2 (see Crossrefs): Sum_{i=0..n} (b(n)+i)^2 = (Sum_{i=n+1..2*n} (b(n)+i)^2) + h*(h-4)*A000217(n)^2 for n>0. - Bruno Berselli, Jan 15 2011
Sequence found by reading the line from 0, in the direction 0, 28, ..., and the line from 9, in the direction 9, 57, ..., in the square spiral whose vertices are the generalized 12-gonal numbers A195162. - Omar E. Pol, Jul 24 2012
Bisection of A195162. - Omar E. Pol, Aug 04 2012

Ivan Panchenko, Table of n, a(n) for n = 0..1000
L. Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
Leo Tavares, Illustration: Diamond Clipped Stars
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Second n-gonal numbers: A005449, A014105, A147875, A045944, A179986, A033954, A062728, this sequence.
Cf. A003154, A000290.
Cf. A195162.

List([0..50], n-> n*(5*n+4)); # G. C. Greubel, Jul 04 2019

[n*(5*n+4): n in [0..50]]; // G. C. Greubel, Jul 04 2019

LinearRecurrence[{3,-3,1}, {0,9,28}, 50] (* or *) Table[5*n^2 + 4*n, {n,0,50}] (* G. C. Greubel, Oct 29 2016 *)
Table[10 Binomial[n,2]+9n,{n,0,60}] (* Harvey P. Dale, Jun 14 2023 *)

a(n) = 10*binomial(n,2) + 9*n \\ Charles R Greathouse IV, Jun 11 2015

[n*(5*n+4) for n in (0..50)] # G. C. Greubel, Jul 04 2019

From R. J. Mathar, Mar 06 2008: (Start)
O.g.f.: x*(9+x)/(1-x)^3.
a(n) = n*(5*n+4). (End)
a(n) = a(n-1) + 10*n - 1 (with a(0)=0). - Vincenzo Librandi, Nov 24 2009
a(n) = Sum_{i=0..n-1} A017377(i) for n>0. - Bruno Berselli, Jan 15 2011
a(n) = A131242(10n+8). - Philippe Deléham, Mar 27 2013
Sum_{n>=1} 1/a(n) = 5/16 + sqrt(1 + 2/sqrt(5))*Pi/8 - 5*log(5)/16 - sqrt(5)*log((1 + sqrt(5))/2)/8 = 0.2155517745488486003038... . - Vaclav Kotesovec, Apr 27 2016
From G. C. Greubel, Oct 29 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: x*(9 + 5*x)*exp(x). (End)
a(n) = A003154(n+1) - A000290(n+1). - Leo Tavares, Mar 29 2022

A218471 a(n) = n(7n-3)/2.

Original entry on oeis.org

0, 2, 11, 27, 50, 80, 117, 161, 212, 270, 335, 407, 486, 572, 665, 765, 872, 986, 1107, 1235, 1370, 1512, 1661, 1817, 1980, 2150, 2327, 2511, 2702, 2900, 3105, 3317, 3536, 3762, 3995, 4235, 4482, 4736, 4997, 5265, 5540, 5822, 6111, 6407, 6710, 7020, 7337

Offset: 0

Philippe Deléham, Mar 26 2013

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. A001106, A022264.

Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488 (this sequence is the case k=7). - Bruno Berselli, Jun 10 2013

List([0..50], n-> n*(7*n-3)/2); # G. C. Greubel, Aug 31 2019

[n*(7*n-3)/2: n in [0..50]]; // G. C. Greubel, Aug 31 2019

seq(n*(7*n-3)/2, n=0..50); # G. C. Greubel, Aug 31 2019

Table[n*(7*n-3)/2, {n,0,50}] (* G. C. Greubel, Aug 23 2017 *)

a(n)=n*(7*n-3)/2 \\ Charles R Greathouse IV, Jun 17 2017

[n*(7*n-3)/2 for n in (0..50)] # G. C. Greubel, Aug 31 2019

G.f.: x*(2+5*x)/(1-x)^3.
a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) with a(0)=0, a(1)=2, a(2)=11.
a(n) = A001106(n) +  n.
a(n) = A022264(n) -  n.
a(n) = A022265(n) - 2*n.
a(n) = A186029(n) - 3*n.
a(n) = A179986(n) - 4*n.
a(n) = A024966(n) - 5*n.
a(n) = A174738(7*n+1).
E.g.f.: (x/2)*(7*x + 4)*exp(x). - G. C. Greubel, Aug 23 2017

A211013 Second 13-gonal numbers: a(n) = n(11n+9)/2.

Original entry on oeis.org

0, 10, 31, 63, 106, 160, 225, 301, 388, 486, 595, 715, 846, 988, 1141, 1305, 1480, 1666, 1863, 2071, 2290, 2520, 2761, 3013, 3276, 3550, 3835, 4131, 4438, 4756, 5085, 5425, 5776, 6138, 6511, 6895, 7290, 7696, 8113, 8541, 8980, 9430, 9891, 10363

Offset: 0

Omar E. Pol, Aug 04 2012

Sequence found by reading the line from 0, in the direction 0, 31... and the line from 10, in the direction 10, 63,..., in the square spiral whose vertices are the generalized 13-gonal numbers A195313.

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

Bisection of A195313.
Second k-gonal numbers (k=5..14): A005449, A014105, A147875, A045944, A179986, A033954, A062728, A135705, this sequence, A211014.
Cf. A051865.

List([0..50], n-> n*(11*n+9)/2); # G. C. Greubel, Jul 04 2019

[n*(11*n+9)/2: n in [0..50]]; // G. C. Greubel, Jul 04 2019

Table[n*(11*n+9)/2, {n,0,50}] (* G. C. Greubel, Jul 04 2019 *)

a(n)=n*(11*n+9)/2 \\ Charles R Greathouse IV, Jun 17 2017

[n*(11*n+9)/2 for n in (0..50)] # G. C. Greubel, Jul 04 2019

G.f.: x*(10+x)/(1-x)^3. - Philippe Deléham, Mar 27 2013
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0) = 0, a(1) = 10, a(2) = 31. - Philippe Deléham, Mar 27 2013
a(n) = A051865(n) + 9n = A180223(n) + 8n = A022268(n) + 5n = A022269(n) + 4n = A152740(n) - n. - Philippe Deléham, Mar 27 2013
a(n) = A218530(11n+9). - Philippe Deléham, Mar 27 2013
E.g.f.: x*(20 + 11*x)*exp(x)/2. - G. C. Greubel, Jul 04 2019

A211014 Second 14-gonal numbers: n(6n+5).

Original entry on oeis.org

0, 11, 34, 69, 116, 175, 246, 329, 424, 531, 650, 781, 924, 1079, 1246, 1425, 1616, 1819, 2034, 2261, 2500, 2751, 3014, 3289, 3576, 3875, 4186, 4509, 4844, 5191, 5550, 5921, 6304, 6699, 7106, 7525, 7956, 8399, 8854, 9321, 9800, 10291, 10794, 11309, 11836, 12375

Offset: 0

Omar E. Pol, Aug 04 2012

Sequence found by reading the line from 0, in the direction 0, 34, ... and the line from 11 in the direction 11, 69, ..., in the square spiral whose vertices are the generalized 14-gonal numbers A195818.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Mark W. Coffey, Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers, arXiv:1601.01673 [math.NT], 2016. See 3rd formula in Proposition 3 p. 36 giving a(n+1).
Leo Tavares, Star illustration
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195818.
Second k-gonal numbers (k=5..14): A005449, A014105, A147875, A045944, A179986, A033954, A062728, A135705, A211013, this sequence.
Cf. A051866.
Cf. A003154.

List([0..50], n-> n*(6*n+5)); # G. C. Greubel, Jul 04 2019

[n*(6*n+5): n in [0..50]]; // G. C. Greubel, Jul 04 2019

Table[n*(6*n+5), {n,0,50}] (* G. C. Greubel, Jul 04 2019 *)
LinearRecurrence[{3,-3,1},{0,11,34},50] (* Harvey P. Dale, Dec 12 2022 *)

a(n)=n*(6*n+5) \\ Charles R Greathouse IV, Jun 17 2017

[n*(6*n+5) for n in (0..50)] # G. C. Greubel, Jul 04 2019

a(n) = -2*Sum_{k=0..n-1} binomial(6*n+5, 6*k+8)*Bernoulli(6*k+8). - Michel Marcus, Jan 11 2016
From G. C. Greubel, Jul 04 2019: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(11+x)/(1-x)^3.
E.g.f.: x*(11+6*x)*exp(x). (End)
From Amiram Eldar, Feb 28 2022: (Start)
Sum_{n>=1} 1/a(n) = sqrt(3)*Pi/10 + 6/25 - 3*log(3)/10 - 2*log(2)/5.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/5 + log(2)/5 - 6/25 - sqrt(3)*log(sqrt(3)+2)/5. (End)
a(n) = A003154(n+1) - n - 1. - Leo Tavares, Jan 29 2023

A195027 a(n) = 2n(7*n + 5).

Original entry on oeis.org

0, 24, 76, 156, 264, 400, 564, 756, 976, 1224, 1500, 1804, 2136, 2496, 2884, 3300, 3744, 4216, 4716, 5244, 5800, 6384, 6996, 7636, 8304, 9000, 9724, 10476, 11256, 12064, 12900, 13764, 14656, 15576, 16524, 17500, 18504, 19536, 20596, 21684, 22800, 23944, 25116, 26316

Offset: 0

Omar E. Pol, Oct 13 2011

Sequence found by reading the line from 0, in the direction 0, 24, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. Semi-axis opposite to A195023 in the same square spiral, which is related to the primitive Pythagorean triple [3, 4, 5].

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A144555, A152760, A179986, A195019, A195020, A195021, A195023, A195024, A195025, A195026, A195320.

Cf. A185019, A193053, A198017.

[14*n^2 +10*n: n in [0..50]]; // Vincenzo Librandi, Oct 14 2011

Table[2n(7n+5),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,24,76},50] (* Harvey P. Dale, Jul 24 2012 *)

a(n)=2*n*(7*n+5) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 14*n^2 + 10*n.
a(n) = 4*A179986(n). - Bruno Berselli, Oct 13 2011
G.f.: 4*x*(6+x)/(1-x)^3. - Colin Barker, Jan 09 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=24, a(2)=76. - Harvey P. Dale, Jul 24 2012
E.g.f.: 2*exp(x)*x*(12 + 7*x). - Elmo R. Oliveira, Dec 30 2024

A255687 a(n) = n(n + 1)(7*n + 11)/6.

Original entry on oeis.org

0, 6, 25, 64, 130, 230, 371, 560, 804, 1110, 1485, 1936, 2470, 3094, 3815, 4640, 5576, 6630, 7809, 9120, 10570, 12166, 13915, 15824, 17900, 20150, 22581, 25200, 28014, 31030, 34255, 37696, 41360, 45254, 49385, 53760, 58386, 63270, 68419, 73840, 79540, 85526

Offset: 0

Luce ETIENNE, Mar 02 2015

This sequence gives the number of triangles of all sizes in (3*n^2+2*n)-polyiamonds in a pentagonal or heptagonal configuration.

Also sum of 2*n*(n+1)*(n+2)/3 triangles oriented in one direction and n*(n+1)^2/2 oriented in the opposite direction.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

First bisection of A212977.
Partial sums of A179986.
Cf. A000292, A006002, A007584, A255211.

[n*(n+1)*(7*n+11)/6: n in [0..50]]; // Bruno Berselli, Mar 02 2015

A255687:=n->n*(n+1)*(7*n+11)/6: seq(A255687(n), n=0..50); # Wesley Ivan Hurt, Mar 03 2015

Table[n (n + 1) (7 n + 11)/6, {n, 0, 50}] (* Bruno Berselli, Mar 02 2015 *)
LinearRecurrence[{4,-6,4,-1},{0,6,25,64},50] (* Harvey P. Dale, Jul 17 2015 *)

PARI
```
vector(50, n, n--; n*(n+1)*(7*n+11)/6)
```

concat(0, Vec(x*(x+6)/(x-1)^4 + O(x^100))) \\ Colin Barker, Mar 02 2015

[n*(n+1)*(7*n+11)/6 for n in (0..50)] # Bruno Berselli, Mar 02 2015

a(n) = (1/2)*(Sum_{j=0..n} (n+1-j)*(3*n-j) + Sum_{j=0..n-1} (n-j)*(3*n+1-3*j)).
From Colin Barker, Mar 02 2015: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: x*(x + 6)/(x - 1)^4. (End)
a(n) = -A007584(-n-1). - Bruno Berselli, Mar 02 2015
From Elmo R. Oliveira, Aug 18 2025: (Start)
E.g.f.: exp(x)*x*(36 + 39*x + 7*x^2)/6.
a(n) = A212977(2*n). (End)

A341768 a(n) = n * (binomial(n,2) - 2).

Original entry on oeis.org

0, -2, -2, 3, 16, 40, 78, 133, 208, 306, 430, 583, 768, 988, 1246, 1545, 1888, 2278, 2718, 3211, 3760, 4368, 5038, 5773, 6576, 7450, 8398, 9423, 10528, 11716, 12990, 14353, 15808, 17358, 19006, 20755, 22608, 24568, 26638, 28821, 31120, 33538, 36078, 38743, 41536, 44460

Offset: 0

Ilya Gutkovskiy, Feb 19 2021

The n-th second n-gonal number.

			a(7) = A147875(7) = A000566(-7) = 133.

Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005449, A005564, A006002, A014105, A033954, A034856, A045944, A060354, A062728, A135705, A147875, A179986, A292551.

Table[n (Binomial[n, 2] - 2), {n, 0, 45}]
LinearRecurrence[{4, -6, 4, -1}, {0, -2, -2, 3}, 46]
CoefficientList[Series[-x (2 - 6 x + x^2)/(1 - x)^4, {x, 0, 45}], x]

G.f.: -x*(2 - 6*x + x^2)/(1 - x)^4.
E.g.f.: -exp(x)*x*(4 - 2*x - x^2)/2.
a(n) = n^2*(n - 1)/2 - 2*n.

cp's OEIS Frontend

A135705 a(n) = 10*binomial(n,2) + 9*n.

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A218471 a(n) = n*(7*n-3)/2.

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A211013 Second 13-gonal numbers: a(n) = n*(11*n+9)/2.

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A211014 Second 14-gonal numbers: n*(6*n+5).

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A195027 a(n) = 2*n*(7*n + 5).

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A255687 a(n) = n*(n + 1)*(7*n + 11)/6.

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Mathematica

A135705 a(n) = 10binomial(n,2) + 9n.

A218471 a(n) = n(7n-3)/2.

A211013 Second 13-gonal numbers: a(n) = n(11n+9)/2.

A211014 Second 14-gonal numbers: n(6n+5).

A195027 a(n) = 2n(7*n + 5).

A255687 a(n) = n(n + 1)(7*n + 11)/6.