A158187 -id:A158187 - cp's OEIS frontend

A033583 a(n) = 10*n^2.

0, 10, 40, 90, 160, 250, 360, 490, 640, 810, 1000, 1210, 1440, 1690, 1960, 2250, 2560, 2890, 3240, 3610, 4000, 4410, 4840, 5290, 5760, 6250, 6760, 7290, 7840, 8410, 9000, 9610, 10240, 10890, 11560, 12250, 12960, 13690, 14440, 15210, 16000, 16810

Offset: 0

N. J. A. Sloane

Number of edges of a complete 5-partite graph of order 5n, K_n,n,n,n,n. - Roberto E. Martinez II, Oct 18 2001
10 times the squares. - Omar E. Pol, Dec 13 2008
Sequence found by reading the line from 0, in the direction 0, 10, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Sep 10 2011

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

Cf. A000217, A000290, A001105, A033428, A033429, A033581, A085787, A158187.

seq(10*n^2,n=0..41); # Nathaniel Johnston, Jun 26 2011

10*Range[0,50]^2  (* Harvey P. Dale, Apr 20 2011 *)

a(n)=10*n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 10*A000290(n) = 5*A001105(n) = 2*A033429(n). - Omar E. Pol, Dec 13 2008
a(n) = A158187(n) - 1. - Reinhard Zumkeller, Mar 13 2009
a(n) = 20*n + a(n-1) - 10 for n>0, a(0)=0. - Vincenzo Librandi, Aug 05 2010
a(n) = t(5*n) - 5*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(5*n) - 5*A000217(n). - Bruno Berselli, Aug 31 2017
From Amiram Eldar, Feb 03 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/60.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/120.
Product_{n>=1} (1 + 1/a(n)) = sqrt(10)*sinh(Pi/sqrt(10))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(10)*sin(Pi/sqrt(10))/Pi. (End)
From Stefano Spezia, Jul 06 2021: (Start)
O.g.f.: 10*x*(1 + x)/(1 - x)^3.
E.g.f.: 10*exp(x)*x*(1 + x). (End)

A010010 a(0) = 1, a(n) = 20*n^2 + 2 for n>0.

Original entry on oeis.org

1, 22, 82, 182, 322, 502, 722, 982, 1282, 1622, 2002, 2422, 2882, 3382, 3922, 4502, 5122, 5782, 6482, 7222, 8002, 8822, 9682, 10582, 11522, 12502, 13522, 14582, 15682, 16822, 18002, 19222, 20482, 21782, 23122, 24502, 25922, 27382, 28882, 30422, 32002, 33622

Offset: 0

N. J. A. Sloane

Bruno Berselli, Table of n, a(n) for n = 0..1000
Volodymyr Mazorchuk and Xiaoyu Zhu, Combinatorics of infinite rank module categories over finite dimensional sl_3-modules in Lie-algebraic context, arXiv:2501.00291 [math.RT], 2024. See p. 2.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A206399.

[1] cat [20*n^2 + 2: n in [1..50]]; // Vincenzo Librandi, Aug 03 2015

Join[{1}, 20 Range[41]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {22, 82, 182}, 50]] (* Vincenzo Librandi, Aug 03 2015 *)

a(n) = A033571(n)+A158186(n) = A158187(n)*2 for n>0. - Reinhard Zumkeller, Mar 13 2009
G.f.: (1+x)*(1+18*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*20+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4+sqrt(10)/40*Pi*coth(Pi/sqrt(10)) = 1.0772981051444036327... - R. J. Mathar, May 07 2024
a(n) = A069133(n)+A069133(n+1). - R. J. Mathar, May 07 2024

A158602 a(n) = 40*n^2 + 1.

Original entry on oeis.org

1, 41, 161, 361, 641, 1001, 1441, 1961, 2561, 3241, 4001, 4841, 5761, 6761, 7841, 9001, 10241, 11561, 12961, 14441, 16001, 17641, 19361, 21161, 23041, 25001, 27041, 29161, 31361, 33641, 36001, 38441, 40961, 43561, 46241, 49001, 51841, 54761, 57761, 60841, 64001

Offset: 0

Vincenzo Librandi, Mar 22 2009

The identity (40*n^2 + 1)^2 - (400*n^2 + 20)*(2*n)^2 = 1 can be written as a(n)^2 - A158601(n)*A005843(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A158187, A158601.

I:=[1,41,161]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 16 2012

A158602:=n->40*n^2; seq(A158602(k), k=0..100); # Wesley Ivan Hurt, Sep 27 2013

40*Range[0,40]^2+1 (* or *) LinearRecurrence[{3,-3,1},{1,41,161},40] (* Harvey P. Dale, Jul 25 2011 *)
Table[40n^2+1, {n,0,100}] (* Wesley Ivan Hurt, Sep 27 2013 *)

for(n=0, 40, print1(40*n^2 + 1", ")); \\ Vincenzo Librandi, Feb 16 2012

G.f.: -(1 + 38*x + 41*x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 16 2023: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi/(2*sqrt(10)))*Pi/(2*sqrt(10)) + 1)/2.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(2*sqrt(10)))*Pi/(2*sqrt(10)) + 1)/2. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: exp(x)*(1 + 40*x + 40*x^2).
a(n) = A158187(2*n). (End)

Comment rewritten, formula replaced by R. J. Mathar, Oct 28 2009

A158445 a(n) = 25*n^2 + 5.

Original entry on oeis.org

30, 105, 230, 405, 630, 905, 1230, 1605, 2030, 2505, 3030, 3605, 4230, 4905, 5630, 6405, 7230, 8105, 9030, 10005, 11030, 12105, 13230, 14405, 15630, 16905, 18230, 19605, 21030, 22505, 24030, 25605, 27230, 28905, 30630, 32405, 34230, 36105, 38030, 40005, 42030

Offset: 1

Vincenzo Librandi, Mar 19 2009

The identity (10*n^2 + 1)^2 - (25*n^2 + 5)*(2*n)^2 = 1 can be written as A158187(n)^2 - a(n)*A005843(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A158187, A212656.

I:=[30, 105, 230]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];

Table[25n^2+5,{n,50}]
LinearRecurrence[{3,-3,1},{30,105,230},50] (* Harvey P. Dale, Mar 21 2025 *)

PARI
```
a(n) = 25*n^2 + 5.
```

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f: 5*x*(6+3*x+x^2)/(1-x)^3.
From Amiram Eldar, Mar 05 2023: (Start)
Sum_{n>=1} 1/a(n) = (coth(Pi/sqrt(5))*Pi/sqrt(5) - 1)/10.
Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/sqrt(5))*Pi/sqrt(5))/10. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 5*(exp(x)*(5*x^2 + 5*x + 1) - 1).
a(n) = 5*A212656(n). (End)

A158492 a(n) = 100*n^2 + 10.

Original entry on oeis.org

10, 110, 410, 910, 1610, 2510, 3610, 4910, 6410, 8110, 10010, 12110, 14410, 16910, 19610, 22510, 25610, 28910, 32410, 36110, 40010, 44110, 48410, 52910, 57610, 62510, 67610, 72910, 78410, 84110, 90010, 96110, 102410, 108910, 115610, 122510, 129610, 136910, 144410

Offset: 0

Vincenzo Librandi, Mar 21 2009

The identity (20*n^2 + 1)^2 - (100*n^2 + 10)*(2*n)^2 = 1 can be written as A158493(n)^2 - a(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 21 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A158187, A158493.

I:=[10, 110, 410]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 21 2012

LinearRecurrence[{3, -3, 1}, {10, 110, 410}, 50] (* Vincenzo Librandi, Feb 21 2012 *)
100Range[0,40]^2+10 (* Harvey P. Dale, Dec 30 2019 *)

for(n=0, 40, print1(100*n^2 + 10", ")); \\ Vincenzo Librandi, Feb 21 2012

From Vincenzo Librandi, Feb 21 2012: (Start)
G.f.: -(10 + 80*x + 110*x^2)/(x-1)^3;
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Mar 05 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + coth(Pi/sqrt(10))*Pi/sqrt(10))/20.
Sum_{n>=0} (-1)^n/a(n) = (1 + cosech(Pi/sqrt(10))*Pi/sqrt(10))/20. (End)
From Elmo R. Oliveira, Jan 17 2025: (Start)
E.g.f.: 10*exp(x)*(1 + 10*x + 10*x^2).
a(n) = 10*A158187(n). (End)

Edited by N. J. A. Sloane, Oct 12 2009

A272130 a(n) = 16n^3 + 10n^2 + 4*n + 1.

Original entry on oeis.org

1, 31, 177, 535, 1201, 2271, 3841, 6007, 8865, 12511, 17041, 22551, 29137, 36895, 45921, 56311, 68161, 81567, 96625, 113431, 132081, 152671, 175297, 200055, 227041, 256351, 288081, 322327, 359185, 398751, 441121, 486391, 534657, 586015, 640561, 698391

Offset: 0

Vincenzo Librandi, Apr 21 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Beck, J. A. De Loera, M. Develin, J. Pfeifle and R. P. Stanley, Coefficients and roots of Ehrhart Polynomials, Contemp. Math. 374 (2005), 15-36, page 19.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A144965, A158187, A272039.

Magma
```
[16*n^3+10*n^2+4*n+1: n in [0..50]];
```

A272130:=n->16*n^3+10*n^2+4*n+1: seq(A272130(n), n=0..50); # Wesley Ivan Hurt, Apr 22 2016

LinearRecurrence[{4,-6,4,-1},{1,31,177,535},50]
CoefficientList[Series[(1 + 27*x + 59*x^2 + 9*x^3)/(1 - x)^4, {x, 0, 30}], x] (* Wesley Ivan Hurt, Apr 22 2016 *)

vector(100, n, n--; 16*n^3+10*n^2+4*n+1) \\ Altug Alkan, Apr 22 2016

O.g.f.: (1+27*x+59*x^2+9*x^3)/(1-x)^4.
E.g.f.: (1+30*x+58*x^2+16*x^3)*exp(x).
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3.
a(n) = A158187(n) + A144965(n).

cp's OEIS Frontend

A033583 a(n) = 10*n^2.

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A010010 a(0) = 1, a(n) = 20*n^2 + 2 for n>0.

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A158602 a(n) = 40*n^2 + 1.

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A158445 a(n) = 25*n^2 + 5.

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Magma

Mathematica

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A158492 a(n) = 100*n^2 + 10.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A272130 a(n) = 16*n^3 + 10*n^2 + 4*n + 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A272130 a(n) = 16n^3 + 10n^2 + 4*n + 1.