A161587 -id:A161587 - cp's OEIS frontend

A161935 28-gonal numbers: a(n) = n(13n - 12).

0, 1, 28, 81, 160, 265, 396, 553, 736, 945, 1180, 1441, 1728, 2041, 2380, 2745, 3136, 3553, 3996, 4465, 4960, 5481, 6028, 6601, 7200, 7825, 8476, 9153, 9856, 10585, 11340, 12121, 12928, 13761, 14620, 15505, 16416, 17353, 18316, 19305, 20320, 21361, 22428, 23521

Offset: 0

Pierre Gayet, Jun 22 2009

The defining formula can be regarded as an approximation and simplification of the expansion / propagation of native hydrophytes on the surface of stagnant waters in orthogonal directions; absence of competition / concurrence and of retrogression is assumed, mortality is taken into account. - [Translation of a comment in French sent by Pierre Gayet]

These are also the star 14-gonal numbers: a(n) = A051866(n) + 14*A000217(n-1). Luciano Ancora, Apr 04 2015

			G.f. = x + 28*x^2 + 81*x^3 + 160*x^4 + 265*x^5 + 396*x^6 + 553*x^7 + ...

Daniel Mondot, Table of n, a(n) for n = 0..1000
Pierre Gayet, Note et Compte rendu (gif version).
Pierre Gayet, Note et Compte Rendu (pdf version).
Pierre Gayet, 98 séquences générées ... par la formule générale indiquée.
Claude Monet, Nymphéas.
Index to sequences related to polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A051866, A161532, A161549, A161587, A161617, A162316.

Magma
```
[ (n+1)*(13*n+1): n in[0..50] ];
```

lst={}; Do[a=13*n^2+14*n+1; AppendTo[lst, a], {n, 0, 5!}]; lst
Table[n*(13*n - 12), {n, 0, 100}] (* Robert Price, Oct 11 2018 *)

{a(n) = n*(13*n - 12)}; /* Michael Somos, Dec 07 2016 */

a(n+1) = a(n) + 26*n + 1. - Vincenzo Librandi, Nov 30 2010
a(n) = A000217(n) + 25*A000217(n-1). - Luciano Ancora, Apr 04 2015
Product_{n>=2} (1 - 1/a(n)) = 13/14. - Amiram Eldar, Jan 22 2021
E.g.f.: exp(x)*(x + 13*x^2). - Nikolaos Pantelidis, Feb 05 2023
From Elmo R. Oliveira, Dec 14 2024: (Start)
G.f.: x*(1 + 25*x)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

Edited by N. J. A. Sloane, Dec 07 2016 at the suggestion of Daniel Sterman.

Definition simplified by Omar E. Pol, Aug 10 2018

A161532 a(n) = 2n^2 + 8n + 1.

Original entry on oeis.org

1, 11, 25, 43, 65, 91, 121, 155, 193, 235, 281, 331, 385, 443, 505, 571, 641, 715, 793, 875, 961, 1051, 1145, 1243, 1345, 1451, 1561, 1675, 1793, 1915, 2041, 2171, 2305, 2443, 2585, 2731, 2881, 3035, 3193, 3355, 3521, 3691, 3865, 4043, 4225, 4411, 4601, 4795, 4993

Offset: 0

Pierre Gayet, Jun 13 2009

The defining formula can be regarded as an approximation and simplification of the expansion / propagation of native hydrophytes on the surface of stagnant waters in orthogonal directions; absence of competition / concurrence and of retrogression is assumed, mortality is taken into account. - [Translation of a comment in French sent by Pierre Gayet]

Numbers of the form 2*n^2 - 7. - Boris Putievskiy, Feb 04 2013

Pierre Gayet, Table of n, a(n) for n = 0..10000
Pierre Gayet, Note et Compte rendu (gif version).
Pierre Gayet, Note et Compte Rendu (pdf version).
Pierre Gayet, 98 séquences générées ... par la formule générale indiquée.
Claude Monet, Nymphéas.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A161549, A161587, A161617, A161935, A162316.

Magma
```
[ 2*n^2+8*n+1: n in [0..50] ];
```

Table[2*n^2+8*n+1, {n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Jun 13 2009 *)
CoefficientList[Series[(1 + 8*x - 5*x^2)/(1-x)^3, {x, 0, 60}], x] (* Vincenzo Librandi, Feb 07 2013 *)

a(n)=2*n^2+8*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = a(n-1) + 4*n + 6 (with a(0)=1). - Vincenzo Librandi, Nov 30 2010
G.f.: (1 + 8*x - 5*x^2)/(1 - x)^3. - Vincenzo Librandi, Feb 07 2013
From Elmo R. Oliveira, Oct 27 2024: (Start)
E.g.f.: (1 + 10*x + 2*x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

More terms from Vladimir Joseph Stephan Orlovsky, Jun 13 2009

A161549 a(n) = 2n^2 + 14n + 1.

Original entry on oeis.org

1, 17, 37, 61, 89, 121, 157, 197, 241, 289, 341, 397, 457, 521, 589, 661, 737, 817, 901, 989, 1081, 1177, 1277, 1381, 1489, 1601, 1717, 1837, 1961, 2089, 2221, 2357, 2497, 2641, 2789, 2941, 3097, 3257, 3421, 3589, 3761, 3937, 4117, 4301, 4489, 4681, 4877, 5077

Offset: 0

Pierre Gayet, Jun 13 2009

The defining formula can be regarded as an approximation and simplification of the expansion/propagation of native hydrophytes on the surface of stagnant waters in orthogonal directions; absence of competition/concurrence and of retrogression is assumed, mortality is taken into account. - [Translation of a comment in French sent by Pierre Gayet]

Pierre Gayet, Table of n, a(n) for n = 0..9999
Pierre Gayet, Note et Compte rendu (gif version).
Pierre Gayet, Note et Compte Rendu (pdf version).
Pierre Gayet, 98 séquences générées ... par la formule générale indiquée.
Claude Monet, Nymphéas.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A161532, A161587, A161617, A161935, A162316.

Magma
```
[ 2*n^2+14*n+1: n in [0..50] ];
```

lst={}; Do[a=2*n^2+14*n+1; AppendTo[lst, a], {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jun 13 2009 *)
CoefficientList[Series[(1 + 14 x - 11 x^2) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Nov 08 2014 *)
Table[2n^2+14n+1,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,17,37},50] (* Harvey P. Dale, Jul 14 2018 *)

Vec((1+14*x-11*x^2)/(1-x)^3 + O(x^100)) \\ Colin Barker, Nov 08 2014

a(n) = a(n-1) + 4*n + 12 (with a(0)=1). - Vincenzo Librandi, Nov 30 2010
G.f.: (1 + 14*x - 11*x^2)/(1-x)^3. - Vincenzo Librandi, Nov 08 2014
From Elmo R. Oliveira, Oct 25 2024: (Start)
E.g.f.: (1 + 16*x + 2*x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

More terms from Vladimir Joseph Stephan Orlovsky, Jun 13 2009

A161617 a(n) = 8n^2 + 20n + 1.

Original entry on oeis.org

1, 29, 73, 133, 209, 301, 409, 533, 673, 829, 1001, 1189, 1393, 1613, 1849, 2101, 2369, 2653, 2953, 3269, 3601, 3949, 4313, 4693, 5089, 5501, 5929, 6373, 6833, 7309, 7801, 8309, 8833, 9373, 9929, 10501, 11089, 11693, 12313, 12949, 13601, 14269, 14953, 15653, 16369

Offset: 0

Pierre Gayet, Jun 14 2009

The defining formula can be regarded as an approximation and simplification of the expansion / propagation of native hydrophytes on the surface of stagnant waters in orthogonal directions; absence of competition / concurrence and of retrogression is assumed, mortality is taken into account. - (Translation of a comment in French sent by P. Gayet)

Pierre Gayet, Table of n, a(n) for n = 0..9999.
Pierre Gayet, Note et Compte rendu (gif version).
Pierre Gayet, Note et Compte Rendu (pdf version).
Pierre Gayet, 98 séquences générées ... par la formule générale indiquée.
Claude Monet, Nymphéas.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A161532, A161549, A161587, A161935, A162316.

Magma
```
[ 8*n^2+20*n+1: n in [0..50] ];
```

a(n)=8*n^2+20*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = a(n-1) + 16*n + 12 (with a(0)=1). - Vincenzo Librandi, Nov 30 2010
From Elmo R. Oliveira, Oct 22 2024: (Start)
G.f.: (1 + 26*x - 11*x^2)/(1 - x)^3.
E.g.f.: (1 + 28*x + 8*x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A162316 a(n) = 5n^2 + 20n + 1.

Original entry on oeis.org

1, 26, 61, 106, 161, 226, 301, 386, 481, 586, 701, 826, 961, 1106, 1261, 1426, 1601, 1786, 1981, 2186, 2401, 2626, 2861, 3106, 3361, 3626, 3901, 4186, 4481, 4786, 5101, 5426, 5761, 6106, 6461, 6826, 7201, 7586, 7981, 8386, 8801, 9226, 9661, 10106, 10561, 11026

Offset: 0

Pierre Gayet, Jul 01 2009

The defining formula can be regarded as an approximation and simplification of the expansion / propagation of native hydrophytes on the surface of stagnant waters in orthogonal directions; absence of competition / concurrence and of retrogression is assumed, mortality is taken into account. - [Translation of a comment in French sent by Pierre Gayet]

Pierre Gayet, Table of n, a(n) for n = 0..9999
Pierre Gayet, Note et Compte rendu (gif version).
Pierre Gayet, Note et Compte Rendu (pdf version).
Pierre Gayet, 98 séquences générées ... par la formule générale indiquée.
Claude Monet, Nymphéas.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A161532, A161549, A161587, A161617, A161935.

Magma
```
[ 5*n^2+20*n+1: n in [0..50] ];
```

lst={}; Do[a=5*n^2+20*n+1; AppendTo[lst, a], {n, 0, 5!}]; lst
Table[5n^2+20n+1,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{1,26,61},40] (* or *) CoefficientList[Series[(14x^2-23x-1)/(x-1)^3,{x,0,40}],x] (* Harvey P. Dale, May 07 2023 *)

a(n)=5*n^2+20*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = a(n-1) + 10*n + 15 (with a(0)=1). - Vincenzo Librandi, Dec 02 2010
G.f.: (14*x^2 - 23*x - 1)/(x - 1)^3. - Harvey P. Dale, May 07 2023
From Elmo R. Oliveira, Oct 25 2024: (Start)
E.g.f.: (5*x^2 + 25*x + 1)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A202141 a(n) = 13n^2 - 16n + 5.

Original entry on oeis.org

5, 2, 25, 74, 149, 250, 377, 530, 709, 914, 1145, 1402, 1685, 1994, 2329, 2690, 3077, 3490, 3929, 4394, 4885, 5402, 5945, 6514, 7109, 7730, 8377, 9050, 9749, 10474, 11225, 12002, 12805, 13634, 14489, 15370, 16277, 17210, 18169, 19154, 20165, 21202, 22265

Offset: 0

Bruno Berselli, Dec 12 2011

Numbers of the form (r*n - r + 1)^2 + ((r+1)*n - r)^2; in this case, r=2.

Inverse binomial transform of this sequence: 5,-3, 26, 0, 0 (0 continued).

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

Cf. A190816 (r=1), A154355 (r=3), A161587.

Magma
```
[13*n^2-16*n+5: n in [0..42]];
```

A202141:=n->13*n^2-16*n+5: seq(A202141(n), n=0..100); # Wesley Ivan Hurt, Oct 09 2017

Table[13 n^2 - 16 n + 5, {n, 0, 42}]
LinearRecurrence[{3,-3,1},{5,2,25},50] (* Harvey P. Dale, Aug 23 2025 *)

for(n=0, 42, print1(13*n^2-16*n+5", "));

G.f.: (5 - 13*x + 34*x^2)/(1-x)^3.
a(n) = A161587(n-1) + 1 with A161587(-1) = 4.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. - Wesley Ivan Hurt, Oct 09 2017
E.g.f.: (5 - 3*x + 13*x^2)*exp(x). - Elmo R. Oliveira, Oct 20 2024

cp's OEIS Frontend

A161935 28-gonal numbers: a(n) = n*(13*n - 12).

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A161532 a(n) = 2*n^2 + 8*n + 1.

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A161549 a(n) = 2*n^2 + 14*n + 1.

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A161617 a(n) = 8*n^2 + 20*n + 1.

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A162316 a(n) = 5*n^2 + 20*n + 1.

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Magma

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A202141 a(n) = 13*n^2 - 16*n + 5.

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A161935 28-gonal numbers: a(n) = n(13n - 12).

A161532 a(n) = 2n^2 + 8n + 1.

A161549 a(n) = 2n^2 + 14n + 1.

A161617 a(n) = 8n^2 + 20n + 1.

A162316 a(n) = 5n^2 + 20n + 1.

A202141 a(n) = 13n^2 - 16n + 5.