A161549 -id:A161549 - cp's OEIS frontend

A053698 a(n) = n^3 + n^2 + n + 1.

1, 4, 15, 40, 85, 156, 259, 400, 585, 820, 1111, 1464, 1885, 2380, 2955, 3616, 4369, 5220, 6175, 7240, 8421, 9724, 11155, 12720, 14425, 16276, 18279, 20440, 22765, 25260, 27931, 30784, 33825, 37060, 40495, 44136, 47989, 52060, 56355, 60880

Offset: 0

Henry Bottomley, Mar 23 2000

a(n) = 1111 in base n.

n^3 + n^2 + n + 1 = (n^2 + 1)*(n + 1), therefore a(n) is never prime. - Alonso del Arte, Apr 22 2014

			a(2) = 15 because 2^3 + 2^2 + 2 + 1 = 8 + 4 + 2 + 1 = 15.
a(3) = 40 because 3^3 + 3^2 + 3 + 1 = 27 + 9 + 3 + 1 = 40.
a(4) = 85 because 4^3 + 4^2 + 4 + 1 = 64 + 16 + 4 + 1 = 85.
From _Bruno Berselli_, Jan 02 2017: (Start)
The terms of the sequence are provided by the row sums of the following triangle (see the seventh formula in the previous section):
.   1;
.   3,   1;
.   9,   5,   1;
.  19,  13,   7,   1;
.  33,  25,  17,   9,   1;
.  51,  41,  31,  21,  11,   1;
.  73,  61,  49,  37,  25,  13,  1;
.  99,  85,  71,  57,  43,  29, 15,  1;
. 129, 113,  97,  81,  65,  49, 33, 17,  1;
. 163, 145, 127, 109,  91,  73, 55, 37, 19,  1;
. 201, 181, 161, 141, 121, 101, 81, 61, 41, 21, 1;
...
Columns from the first to the fifth, respectively: A058331, A001844, A056220 (after -1), A059993, A161532. Also, eighth column is A161549.
(End)

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

Cf. A237627 (subset of semiprimes).
Cf. A056106 (first differences).
Cf. A062158, A027444, A104878, A055129.

[n^3+n^2+n+1: n in [0..50]]; // Vincenzo Librandi, May 01 2011

A053698:=n->n^3 + n^2 + n + 1; seq(A053698(n), n=0..50); # Wesley Ivan Hurt, Apr 22 2014

Table[n^3 + n^2 + n + 1, {n, 0, 39}] (* Alonso del Arte, Apr 22 2014 *)
FromDigits["1111", Range[0, 50]] (* Paolo Xausa, May 11 2024 *)

Vec((1 + 5*x^2) / (1 - x)^4 + O(x^50)) \\ Colin Barker, Jan 02 2017

def a(n): return (n**3+n**2+n+1) # Torlach Rush, May 08 2024

For n >= 2, a(n) = (n^4-1)/(n-1) = A024002(n)/A024000(n) = A002522(n)*(n+1) = A002061(n+1) + A000578(n).
G.f.: (1+5*x^2) / (1-x)^4. - Colin Barker, Jan 06 2012
a(n) = -A062158(-n). - Bruno Berselli, Jan 26 2016
a(n) = Sum_{i=0..n} 2*n*(n-i)+1. - Bruno Berselli, Jan 02 2017
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3. - Colin Barker, Jan 02 2017
a(n) = A104878(n+3,n) = A055129(4,n) for n > 0. - Mathew Englander, Jan 06 2021
E.g.f.: exp(x)*(x^3+4*x^2+3*x+1). - Nikolaos Pantelidis, Feb 06 2023

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

Original entry on oeis.org

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

A161587 a(n) = 13n^2 + 10n + 1.

Original entry on oeis.org

1, 24, 73, 148, 249, 376, 529, 708, 913, 1144, 1401, 1684, 1993, 2328, 2689, 3076, 3489, 3928, 4393, 4884, 5401, 5944, 6513, 7108, 7729, 8376, 9049, 9748, 10473, 11224, 12001, 12804, 13633, 14488, 15369, 16276, 17209, 18168, 19153, 20164

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 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, A161617, A161935, A162316, A202141.

Magma
```
[ 13*n^2+10*n+1: n in [0..50] ];
```

Table[13n^2+10n+1,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{1,24,73},40] (* Harvey P. Dale, Nov 06 2014 *)

a(n)=13*n^2+10*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = a(n-1) + 26*n - 3 (with a(0)=1). - Vincenzo Librandi, Nov 30 2010
From Bruno Berselli, Dec 12 2011: (Start)
G.f.: (1 + 21*x + 4*x^2)/(1-x)^3.
a(n-1) = A202141(n) - 1 with a(-1)=4. (End)
E.g.f.: exp(x)*(1 + 23*x + 13*x^2). - Stefano Spezia, Oct 21 2024

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)

cp's OEIS Frontend

A053698 a(n) = n^3 + n^2 + n + 1.

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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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A161587 a(n) = 13*n^2 + 10*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

Mathematica

PARI

Formula

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

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

A161587 a(n) = 13n^2 + 10n + 1.

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

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