A069190 -id:A069190 - cp's OEIS frontend

A135453 a(n) = 12*n^2.

0, 12, 48, 108, 192, 300, 432, 588, 768, 972, 1200, 1452, 1728, 2028, 2352, 2700, 3072, 3468, 3888, 4332, 4800, 5292, 5808, 6348, 6912, 7500, 8112, 8748, 9408, 10092, 10800, 11532, 12288, 13068, 13872, 14700, 15552, 16428, 17328, 18252, 19200, 20172, 21168, 22188

Offset: 0

Ben Paul Thurston, Dec 14 2007

Areas of perfect 4:3 rectangles (for n > 0).
Sequence found by reading the line from 0, in the direction 0, 12, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Semi-axis opposite to A069190 in the same spiral. - Omar E. Pol, Sep 16 2011
(x,y,z) = (-a(n), 1 + n*a(n), 1 - n*a(n)) are solutions of the Diophantine equation x^3 + 2*y^3 + 2*z^3 = 4. - XU Pingya, Apr 30 2022

			192 is on the list since 16*12 is a 4:3 rectangle with integer sides and an area of 192.

Ivan Panchenko, Table of n, a(n) for n = 0..10000
John Elias, Illustration: Even Ordered Star Perimeters.
Leo Tavares, Illustration.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A001105, A016742, A033428, A033581, A069190, A086729.

Cf. A008594, A195143.

List([0..100],n->12*n^2); # Muniru A Asiru, Jan 29 2018

seq(12*h^2,n=0..100); # Muniru A Asiru, Jan 29 2018

Table[12*n^2, {n, 0, 60}] (* Stefan Steinerberger, Dec 17 2007 *)
LinearRecurrence[{3,-3,1},{0,12,48},50] (* Harvey P. Dale, Jan 19 2020 *)

a(n)=12*n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 12*A000290(n) = 6*A001105(n) = 4*A033428(n) = 3*A016742(n) = 2*A033581(n). - Omar E. Pol, Dec 13 2008
From Amiram Eldar, Feb 03 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/72 (A086729).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/144.
Product_{n>=1} (1 + 1/a(n)) = 2*sqrt(3)*sinh(Pi/(2*sqrt(3)))/Pi.
Product_{n>=1} (1 - 1/a(n)) = 2*sqrt(3)*sin(Pi/(2*sqrt(3)))/Pi. (End)
From Elmo R. Oliveira, Nov 30 2024: (Start)
G.f.: 12*x*(1 + x)/(1-x)^3.
E.g.f.: 12*x*(1 + x)*exp(x).
a(n) = n*A008594(n) = A195143(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

More terms from Stefan Steinerberger, Dec 17 2007

Minor edits from Omar E. Pol, Dec 15 2008

A069133 Centered 20-gonal (or icosagonal) numbers.

Original entry on oeis.org

1, 21, 61, 121, 201, 301, 421, 561, 721, 901, 1101, 1321, 1561, 1821, 2101, 2401, 2721, 3061, 3421, 3801, 4201, 4621, 5061, 5521, 6001, 6501, 7021, 7561, 8121, 8701, 9301, 9921, 10561, 11221, 11901, 12601, 13321, 14061, 14821, 15601, 16401, 17221, 18061, 18921, 19801

Offset: 1

Terrel Trotter, Jr., Apr 07 2002

Equals binomial transform of [1, 20, 20, 0, 0, 0, ...]. - Gary W. Adamson, Jun 13 2008
Equals Narayana transform (A001263) of [1, 20, 0, 0, 0, ...]. - Gary W. Adamson, Jul 28 2011
Sequence found by reading the line from 1, in the direction 1, 21, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. Semi-axis opposite to A033583 in the same spiral. - Omar E. Pol, Sep 16 2011

			a(5)=201 because 201 = 10*5^2 - 10*5 + 1 = 250 - 50 + 1.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
John Elias, Illustration of initial terms.
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. centered polygonal numbers listed in A069190.

[10*n^2 - 10*n + 1 : n in [1..60]]; // Wesley Ivan Hurt, Oct 10 2021

FoldList[#1 + #2 &, 1, 20 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
Table[10n^2-10n+1,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,21,61}, 50]  (* Harvey P. Dale, Apr 29 2011 *)

a(n)=10*n*(n-1)+1 \\ Charles R Greathouse IV, Jul 29 2011

a(n) = 10n^2 - 10n + 1.
a(n) = 20*n + a(n-1) - 20 with a(1)=1. - Vincenzo Librandi, Aug 08 2010
G.f.: x*(1 + 18*x + x^2)/(1-x)^3. - R. J. Mathar, Feb 04 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=21, a(2)=61. - Harvey P. Dale, Apr 29 2011
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(3/5)*Pi/2)/(2*sqrt(15)).
Sum_{n>=1} a(n)/n! = 11*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 11/e - 1. (End)
a(n) = 12*A000217(n-1) + A016754(n-1). - John Elias, Oct 23 2020
E.g.f.: exp(x)*(1 + 10*x^2) - 1. - Nikolaos Pantelidis, Feb 06 2023

A144129 a(n) = ChebyshevT(3, n).

Original entry on oeis.org

0, 1, 26, 99, 244, 485, 846, 1351, 2024, 2889, 3970, 5291, 6876, 8749, 10934, 13455, 16336, 19601, 23274, 27379, 31940, 36981, 42526, 48599, 55224, 62425, 70226, 78651, 87724, 97469, 107910, 119071, 130976, 143649, 157114, 171395, 186516

Offset: 0

Vladimir Joseph Stephan Orlovsky, Sep 11 2008

The general formula for alternating sums of powers of odd integers is in terms of the Swiss-Knife polynomials P(n,x) A153641 (P(n,0)-(-1)^k*P(n,2*k))/2. Here n=3, thus a(k) = |(P(3,0)-(-1)^k*P(3,2*k))/2|. - Peter Luschny, Jul 12 2009

Partial sums of A069190. - J. M. Bergot, Jul 13 2013

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. A000292, A069190, A153641.

[ 4*n^3-3*n: n in [0..36] ]; // Klaus Brockhaus, Jan 11 2009

I:=[0, 1, 26, 99]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 30 2012

a := n -> (4*n^2-3)*n; # Peter Luschny, Jul 12 2009

lst={};Do[AppendTo[lst,ChebyshevT[3,n]],{n,0,10^2}];lst
Round[Table[N[Cosh[3 ArcCosh[n]], 100], {n, 0, 20}]] (* Artur Jasinski, Feb 14 2010 *)
CoefficientList[Series[x*(1+22*x+x^2)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jun 30 2012 *)
LinearRecurrence[{4,-6,4,-1},{0,1,26,99},40] (* Harvey P. Dale, Apr 02 2015 *)

a(n) = 4*n^3-3*n \\ Charles R Greathouse IV, Feb 08 2012

a(n) = 4*n^3 - 3*n. - Klaus Brockhaus, Jan 11 2009
G.f.: x*(1 + 22*x + x^2)/(1 - x)^4. - Klaus Brockhaus, Jan 11 2009
a(n) = cosh(3*arccosh(n)) = cos(3*arccos(n)). - Artur Jasinski, Feb 14 2010
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 30 2012
a(n) = 24*A000292(n-1) + n. - Bruce J. Nicholson, Jun 12 2020
From Gerry Martens, Apr 06 2024: (Start)
a(n) = Imaginary part of -(1/2)*(2*n*i-1)^3.
a(n) = -4*(1/4 + n^2)^(3/2)*sin(3*arctan(2*n)). (End)

A195143 a(n) = n-th concentric 12-gonal number.

Original entry on oeis.org

0, 1, 12, 25, 48, 73, 108, 145, 192, 241, 300, 361, 432, 505, 588, 673, 768, 865, 972, 1081, 1200, 1321, 1452, 1585, 1728, 1873, 2028, 2185, 2352, 2521, 2700, 2881, 3072, 3265, 3468, 3673, 3888, 4105, 4332, 4561, 4800, 5041, 5292, 5545, 5808, 6073, 6348

Offset: 0

Omar E. Pol, Sep 17 2011

Concentric dodecagonal numbers. [corrected by Ivan Panchenko, Nov 09 2013]
Sequence found by reading the line from 0, in the direction 0, 12,..., and the same line from 1, in the direction 1, 25,..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Main axis, perpendicular to A028896 in the same spiral.
Partial sums of A091998. - Reinhard Zumkeller, Jan 07 2012
Column 12 of A195040. - Omar E. Pol, Sep 28 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

A135453 and A069190 interleaved.
Cf. A001082, A032527, A032528, A077221, A195043, A195045, A195040, A195142, A195145, A195146, A195147, A195148, A195149.
Cf. A016921 (6n+1), A016969 (6n+5), A091998 (positive integers of the form 12*k +- 1), A092242 (positive integers of the form 12*k +- 5).

a195143 n = a195143_list !! n
a195143_list = scanl (+) 0 a091998_list
-- Reinhard Zumkeller, Jan 07 2012

[(3*n^2+(-1)^n-1): n in [0..50]]; // Vincenzo Librandi, Sep 27 2011

Table[Sum[2*(-1)^(n - k + 1) + 6*k - 3, {k, n}], {n, 0, 47}] (* L. Edson Jeffery, Sep 14 2014 *)

From Vincenzo Librandi, Sep 27 2011: (Start)
a(n) = 3*n^2+(-1)^n-1.
a(n) = -a(n-1) + 6*n^2 - 6*n + 1. (End)
G.f.: -x*(1+10*x+x^2) / ( (1+x)*(x-1)^3 ). - R. J. Mathar, Sep 18 2011
a(n) = Sum_{k=1..n} (2*(-1)^(n-k+1) + 3*(2*k-1)), n>0, a(0) = 0. - L. Edson Jeffery, Sep 14 2014
Sum_{n>=1} 1/a(n) = Pi^2/72 + tan(Pi/sqrt(6))*Pi/(4*sqrt(6)). - Amiram Eldar, Jan 16 2023

A010014 a(0) = 1, a(n) = 24*n^2 + 2 for n>0.

Original entry on oeis.org

1, 26, 98, 218, 386, 602, 866, 1178, 1538, 1946, 2402, 2906, 3458, 4058, 4706, 5402, 6146, 6938, 7778, 8666, 9602, 10586, 11618, 12698, 13826, 15002, 16226, 17498, 18818, 20186, 21602, 23066, 24578, 26138, 27746, 29402, 31106, 32858, 34658, 36506, 38402, 40346

Offset: 0

N. J. A. Sloane

Number of points of L_infinity norm n in the simple cubic lattice Z^3. - N. J. A. Sloane, Apr 15 2008
Numbers of cubes needed to completely "cover" another cube. - Xavier Acloque, Oct 20 2003
First bisection of A005897. After 1, all terms are in A000408. - Bruno Berselli, Feb 06 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Xavier Acloque, Polynexus Numbers and other mathematical wonders [broken link]
Victor Kamel, Hanxueyu Yan, and Sean Chester, CLOVER: A GPU-native, Spatio-graph-based Approach to Exact kNN, ACM Int'l Conf. Supercomp. (ICS 2025). See p. 3.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A206399.

Join[{1}, 24 Range[41]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)

a(n) = if (n==0, 1, 24*n^2 + 2);
vector(40, n, a(n-1)) \\ Altug Alkan, Sep 29 2015

a(n) = (2*n+1)^3 - (2*n-1)^3 for n >= 1. - Xavier Acloque, Oct 20 2003
G.f.: (1+x)*(1+22*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
a(n) = (2*n-1)^2 + (2*n+1)^2 + (4*n)^2 for n>0. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*24+2)*exp(x)-1. - Gopinath A. R., Feb 14 2012
a(n) = A005899(n) + A195322(n), n > 0. - R. J. Cano, Sep 29 2015
Sum_{n>=0} 1/a(n) = 3/4 + sqrt(3)/24*Pi*coth(Pi*sqrt(3)/6) = 1.065052868574... - R. J. Mathar, May 07 2024
a(n) = 2*A158480(n), n>0. - R. J. Mathar, May 07 2024
a(n) = A069190(n)+A069190(n+1). - R. J. Mathar, May 07 2024

More terms from Xavier Acloque, Oct 20 2003

A195317 Centered 40-gonal numbers.

Original entry on oeis.org

1, 41, 121, 241, 401, 601, 841, 1121, 1441, 1801, 2201, 2641, 3121, 3641, 4201, 4801, 5441, 6121, 6841, 7601, 8401, 9241, 10121, 11041, 12001, 13001, 14041, 15121, 16241, 17401, 18601, 19841, 21121, 22441, 23801, 25201, 26641, 28121, 29641, 31201, 32801, 34441, 36121

Offset: 1

Omar E. Pol, Sep 16 2011

Also centered tetracontagonal numbers or centered tetrakaicontagonal numbers. Also sequence found by reading the line from 1, in the direction 1, 41, ..., in the square spiral whose vertices are the generalized dodecagonal numbers A195162. Semi-axis opposite to A195322 in the same spiral.

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

Bisection of A195148.
Cf. A003154, A069129, A069133, A069190, A195314, A195315, A195316, A195318.
Cf. A195162, A195322.

[20*n^2 - 20*n + 1: n in [1..50]]; // Vincenzo Librandi, Sep 21 2011

Table[20*n^2 - 20*n + 1, {n, 1, 40}] (* Amiram Eldar, Feb 11 2022 *)

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

a(n) = 20*n^2 - 20*n + 1.
Sum_{n>=1} 1/a(n) = Pi*tan(Pi/sqrt(5))/(8*sqrt(5)). - Amiram Eldar, Feb 11 2022
G.f.: -x*(1+38*x+x^2)/(x-1)^3. - R. J. Mathar, May 07 2024
From Elmo R. Oliveira, Nov 15 2024: (Start)
E.g.f.: exp(x)*(20*x^2 + 1) - 1.
a(n) = 2*A069133(n) - 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3.  (End)

A069131 Centered 18-gonal numbers.

Original entry on oeis.org

1, 19, 55, 109, 181, 271, 379, 505, 649, 811, 991, 1189, 1405, 1639, 1891, 2161, 2449, 2755, 3079, 3421, 3781, 4159, 4555, 4969, 5401, 5851, 6319, 6805, 7309, 7831, 8371, 8929, 9505, 10099, 10711, 11341, 11989, 12655, 13339, 14041, 14761, 15499, 16255, 17029, 17821

Offset: 1

Terrel Trotter, Jr., Apr 07 2002

Equals binomial transform of [1, 18, 18, 0, 0, 0, ...]. Example: a(3) = 55 = (1, 2, 1) dot (1, 18, 18) = (1 + 36 + 18). - Gary W. Adamson, Aug 24 2010
Narayana transform (A001263) of [1, 18, 0, 0, 0, ...]. - Gary W. Adamson, Jul 28 2011
From Lamine Ngom, Aug 19 2021: (Start)
Sequence is a spoke of the hexagonal spiral built from the terms of A016777 (see illustration in links section).
a(n) is a bisection of A195042.
a(n) is a trisection of A028387.
a(n) + 1 is promic (A002378).
a(n) + 2 is a trisection of A002061.
a(n) + 9 is the arithmetic mean of its neighbors.
4*a(n) + 5 is a square: A016945(n)^2. (End)

			a(5) = 181 because 9*5^2 - 9*5 + 1 = 225 - 45 + 1 = 181.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
John Elias, Illustration of Initial Terms: Triangular & Hexagonal Configurations.
Lamine Ngom, An origin of A069131 (illustration).
Leo Tavares, Illustration: Tri-Hexagons.
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. centered polygonal numbers listed in A069190.
Cf. A000217, A028387, A195042, A016945, A002378, A082040, A304163, A003215, A247792, A016777,A016778, A016790, A010008, A008600, A002061.
Cf. A000290, A139278, A069129, A062786, A033996, A060544, A027468, A016754, A124080, A069099, A152740, A049598, A005891, A152741, A001844, A163756, A005448, A194715.

[9*n^2 - 9*n + 1 : n in [1..50]]; // Wesley Ivan Hurt, May 05 2021

FoldList[#1 + #2 &, 1, 18 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
LinearRecurrence[{3,-3,1},{1,19,55},50] (* Harvey P. Dale, Jan 20 2014 *)

a(n)=9*n^2-9*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 9*n^2 - 9*n + 1.
a(n) = 18*n + a(n-1) - 18 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
G.f.: ( x*(1+16*x+x^2) ) / ( (1-x)^3 ). - R. J. Mathar, Feb 04 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=19, a(3)=55. - Harvey P. Dale, Jan 20 2014
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(5)*Pi/6)/(3*sqrt(5)).
Sum_{n>=1} a(n)/n! = 10*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 10/e - 1. (End)
From Lamine Ngom, Aug 19 2021: (Start)
a(n) = 18*A000217(n) + 1 = 9*A002378(n) + 1.
a(n) = 3*A003215(n) - 2.
a(n) = A247792(n) - 9*n.
a(n) = A082040(n) + A304163(n) - a(n-1) = A016778(n) + A016790(n) - a(n-1), n > 0.
a(n) + a(n+1) = 2*A247792(n) = A010008(n), n > 0.
a(n+1) - a(n) = 18*n = A008600(n). (End)
From Leo Tavares, Oct 31 2021: (Start)
a(n)= A000290(n) + A139278(n-1)
a(n) = A069129(n) + A002378(n-1)
a(n) = A062786(n) + 8*A000217(n-1)
a(n) = A062786(n) + A033996(n-1)
a(n) = A060544(n) + 9*A000217(n-1)
a(n) = A060544(n) + A027468(n-1)
a(n) = A016754(n-1) + 10*A000217(n-1)
a(n) = A016754(n-1) + A124080
a(n) = A069099(n) + 11*A000217(n-1)
a(n) = A069099(n) + A152740(n-1)
a(n) = A003215(n-1) + 12*A000217(n-1)
a(n) = A003215(n-1) + A049598(n-1)
a(n) = A005891(n-1) + 13*A000217(n-1)
a(n) = A005891(n-1) + A152741(n-1)
a(n) = A001844(n) + 14*A000217(n-1)
a(n) = A001844(n) + A163756(n-1)
a(n) = A005448(n) + 15*A000217(n-1)
a(n) = A005448(n) + A194715(n-1). (End)
E.g.f.: exp(x)*(1 + 9*x^2) - 1. - Nikolaos Pantelidis, Feb 06 2023

A069173 Centered 22-gonal numbers.

Original entry on oeis.org

1, 23, 67, 133, 221, 331, 463, 617, 793, 991, 1211, 1453, 1717, 2003, 2311, 2641, 2993, 3367, 3763, 4181, 4621, 5083, 5567, 6073, 6601, 7151, 7723, 8317, 8933, 9571, 10231, 10913, 11617, 12343, 13091, 13861, 14653, 15467, 16303, 17161, 18041, 18943, 19867, 20813

Offset: 1

Terrel Trotter, Jr., Apr 09 2002

			a(5) = 221 because 11*5^2 - 11*5 + 1 = 275 - 55 + 1 = 221
For n=2, a(2)=22*2+1-22=23; n=3, a(3)=22*3+23-22=67; n=4, a(4)=22*4+67-22=133.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. centered polygonal numbers listed in A069190.

FoldList[#1 + #2 &, 1, 22 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)

a(n)=11*n^2-11*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 11*n^2 - 11*n + 1.
a(n) = 22*n + a(n-1) - 22 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(7/11)*Pi/2)/sqrt(77).
Sum_{n>=1} a(n)/n! = 12*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 12/e - 1. (End)
E.g.f.: exp(x)*(1 + 11 * x^2) - 1. - Nikolaos Pantelidis, Feb 06 2023
From Elmo R. Oliveira, Oct 22 2024: (Start)
G.f.: x*(1 + 20*x + x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A262221 a(n) = 25n(n + 1)/2 + 1.

Original entry on oeis.org

1, 26, 76, 151, 251, 376, 526, 701, 901, 1126, 1376, 1651, 1951, 2276, 2626, 3001, 3401, 3826, 4276, 4751, 5251, 5776, 6326, 6901, 7501, 8126, 8776, 9451, 10151, 10876, 11626, 12401, 13201, 14026, 14876, 15751, 16651, 17576, 18526, 19501, 20501, 21526, 22576, 23651

Offset: 0

Bruno Berselli, Sep 15 2015

Also centered 25-gonal (or icosipentagonal) numbers.
This is the case k=25 of the formula (k*n*(n+1) - (-1)^k + 1)/2. See table in Links section for similar sequences.
For k=2*n, the formula shown above gives A011379.
Primes in sequence: 151, 251, 701, 1951, 3001, 4751, 10151, 12401, ...

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 51 (23rd row of the table).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Table of numbers of the form (k*n*(n+1)-(-1)^k+1)/2.
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A008607 (first differences), A011379, A034856, A054254, A080956, A123296, A186349, A255184.
Cf. centered polygonal numbers listed in A069190.
Similar sequences of the form (k*n*(n+1) - (-1)^k + 1)/2 with -1 <= k <= 26: A000004, A000124, A002378, A005448, A005891, A028896, A033996, A035008, A046092, A049598, A060544, A064200, A069099, A069125, A069126, A069128, A069130, A069132, A069174, A069178, A080956, A124080, A163756, A163758, A163761, A164136, A173307.

Magma
```
[25*n*(n+1)/2+1: n in [0..50]];
```

Table[25 n (n + 1)/2 + 1, {n, 0, 50}]
25*Accumulate[Range[0,50]]+1 (* or *) LinearRecurrence[{3,-3,1},{1,26,76},50] (* Harvey P. Dale, Jan 29 2023 *)

PARI
```
vector(50, n, n--; 25*n*(n+1)/2+1)
```
Sage
```
[25*n*(n+1)/2+1 for n in (0..50)]
```

G.f.: (1 + 23*x + x^2)/(1 - x)^3.
a(n) = a(-n-1) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = A123296(n) + 1.
a(n) = A000217(5*n+2) - 2.
a(n) = A034856(5*n+1).
a(n) = A186349(10*n+1).
a(n) = A054254(5*n+2) with n>0, a(0)=1.
a(n) = A000217(n+1) + 23*A000217(n) + A000217(n-1) with A000217(-1)=0.
Sum_{i>=0} 1/a(i) = 1.078209111... = 2*Pi*tan(Pi*sqrt(17)/10)/(5*sqrt(17)).
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=0} a(n)/n! = 77*e/2.
Sum_{n>=0} (-1)^(n+1) * a(n)/n! = 23/(2*e). (End)
E.g.f.: exp(x)*(2 + 50*x + 25*x^2)/2. - Elmo R. Oliveira, Dec 24 2024

A069130 Centered 17-gonal numbers: (17n^2 - 17n + 2)/2.

Original entry on oeis.org

1, 18, 52, 103, 171, 256, 358, 477, 613, 766, 936, 1123, 1327, 1548, 1786, 2041, 2313, 2602, 2908, 3231, 3571, 3928, 4302, 4693, 5101, 5526, 5968, 6427, 6903, 7396, 7906, 8433, 8977, 9538, 10116, 10711, 11323, 11952, 12598, 13261, 13941, 14638, 15352

Offset: 1

Terrel Trotter, Jr., Apr 07 2002

Equals binomial transform of [1, 17, 17, 0, 0, 0, ...]. - Gary W. Adamson, Mar 26 2010

			a(5) = 171 because (17*5^2 - 17*5 + 2)/2 = (425 - 85 + 2)/2 = 342/2 = 171.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers
Index entries for sequences related to centered polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1)

Cf. centered polygonal numbers listed in A069190.

[ (17*n^2 - 17*n + 2)/2 : n in [1..50] ]; // Wesley Ivan Hurt, Jun 09 2014

A069130:=n->(17*n^2 - 17*n + 2)/2; seq(A069130(n), n=1..50); # Wesley Ivan Hurt, Jun 09 2014

FoldList[#1 + #2 &, 1, 17 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
Table[(17n^2-17n+2)/2,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,18,52},50] (* Harvey P. Dale, Jun 05 2011 *)

a(n)=17*binomial(n,2)+1 \\ Charles R Greathouse IV, Jun 05 2011

a(n) = (17*n^2 - 17*n + 2)/2.
a(n) = 17*n + a(n-1) - 17 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
G.f.: x*(1+15*x+x^2) / (1-x)^3. - R. J. Mathar, Feb 04 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=18, a(2)=52. - Harvey P. Dale, Jun 05 2011
Narayana transform (A001263) of [1, 17, 0, 0, 0, ...]. - Gary W. Adamson, Jul 28 2011
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = 2*Pi*tan(3*Pi/(2*sqrt(17)))/(3*sqrt(17)).
Sum_{n>=1} a(n)/n! = 19*e/2 - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 19/(2*e) - 1. (End)
E.g.f.: exp(x)*(1 + 17*x^2/2) - 1. - Stefano Spezia, May 31 2022

Typo in formula fixed by Omar E. Pol, Dec 22 2008

cp's OEIS Frontend

A135453 a(n) = 12*n^2.

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A069133 Centered 20-gonal (or icosagonal) numbers.

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A144129 a(n) = ChebyshevT(3, n).

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A195143 a(n) = n-th concentric 12-gonal number.

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A010014 a(0) = 1, a(n) = 24*n^2 + 2 for n>0.

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A195317 Centered 40-gonal numbers.

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A069131 Centered 18-gonal numbers.

A262221 a(n) = 25n(n + 1)/2 + 1.

A069130 Centered 17-gonal numbers: (17n^2 - 17n + 2)/2.