A135453 -id:A135453 - cp's OEIS frontend

A069190 Centered 24-gonal numbers.

1, 25, 73, 145, 241, 361, 505, 673, 865, 1081, 1321, 1585, 1873, 2185, 2521, 2881, 3265, 3673, 4105, 4561, 5041, 5545, 6073, 6625, 7201, 7801, 8425, 9073, 9745, 10441, 11161, 11905, 12673, 13465, 14281, 15121, 15985, 16873, 17785, 18721, 19681, 20665, 21673

Offset: 1

Terrel Trotter, Jr., Apr 10 2002

Sequence found by reading the line from 1, in the direction 1, 25, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Semi-axis opposite to A135453 in the same spiral. - Omar E. Pol, Sep 16 2011

			a(5) = 241 because 12*5^2 - 12*5 + 1 = 300 - 60 + 1 = 241.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
John Elias, Illustration: Odd Ordered Star Perimeters.
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 k-gonal numbers with k=3..25: A005448, A001844, A005891, A003215, A069099, A016754, A060544, A062786, A069125, A003154, A069126, A069127, A069128, A069129, A069130, A069131, A069132, A069133, A069178, A069173, A069174, A069190, A262221.

FoldList[#1 + #2 &, 1, 24 Range@ 45] (* Robert G. Wilson v *)
Table[12n^2-12n+1,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,25,73},50] (* Harvey P. Dale, Jul 17 2011 *)

a(n)=12*n^2-12*n+1 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 12*n^2 - 12*n + 1.
a(n) = 24*n + a(n-1) - 24 with a(1)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=25, a(3)=73. - Harvey P. Dale, Jul 17 2011
G.f.: x*(1+22*x+x^2)/(1-x)^3. - Harvey P. Dale, Jul 17 2011
Binomial transform of [1, 24, 24, 0, 0, 0, ...] and Narayana transform (cf. A001263) of [1, 24, 0, 0, 0, ...]. - Gary W. Adamson, Jul 26 2011
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi*tan(Pi/sqrt(6))/(4*sqrt(6)).
Sum_{n>=1} a(n)/n! = 13*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 13/e - 1. (End)
E.g.f.: exp(x)*(1 + 12*x^2) - 1. - Stefano Spezia, May 31 2022

More terms from Harvey P. Dale, Jul 17 2011

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

A244630 a(n) = 17*n^2.

Original entry on oeis.org

0, 17, 68, 153, 272, 425, 612, 833, 1088, 1377, 1700, 2057, 2448, 2873, 3332, 3825, 4352, 4913, 5508, 6137, 6800, 7497, 8228, 8993, 9792, 10625, 11492, 12393, 13328, 14297, 15300, 16337, 17408, 18513, 19652, 20825, 22032, 23273, 24548, 25857, 27200, 28577, 29988

Offset: 0

Vincenzo Librandi, Jul 03 2014

First bisection of A195047. - Bruno Berselli, Jul 03 2014

Norms of purely imaginary numbers in Z[sqrt(-17)] (for example, 3*sqrt(-17) has norm 153). - Alonso del Arte, Jun 23 2018

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

Cf. A008599, A195047.

Cf. similar sequences of the type k*n^2: A000290 (k = 1), A001105 (k = 2), A033428 (k = 3), A016742 (k = 4), A033429 (k = 5), A033581 (k = 6), A033582 (k = 7), A139098 (k = 8), A016766 (k = 9), A033583 (k = 10), A033584 (k = 11), A135453 (k = 12), A152742 (k = 13), A144555 (k = 14), A064761 (k = 15), A016802 (k = 16), this sequence (k = 17), A195321 (k = 18), A244631 (k = 19), A195322 (k = 20), A064762 (k = 21), A195323 (k = 22), A244632 (k = 23), A195824 (k = 24), A016850 (k = 25), A244633 (k = 26), A244634 (k = 27), A064763 (k = 28), A244635 (k = 29), A244636 (k = 30).

List([0..45],n->17*n^2); # Muniru A Asiru, Jun 29 2018

Magma
```
[17*n^2: n in [0..40]];
```

seq(17*n^2,n=0..45); # Muniru A Asiru, Jun 29 2018

Mathematica
```
Table[17 n^2, {n, 0, 40}]
```

a(n)=17*n^2 \\ Charles R Greathouse IV, Oct 07 2015

for (i <- 0 to 50) yield 17 * (i * i) // Alonso del Arte, Jun 29 2018

G.f.: 17*x*(1 + x)/(1 - x)^3. [corrected by Bruno Berselli, Jul 03 2014]
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = 17*A000290(n). - Omar E. Pol, Jul 03 2014
a(n) = a(-n). - Muniru A Asiru, Jun 29 2018
From Elmo R. Oliveira, Dec 02 2024: (Start)
E.g.f.: 17*x*(1 + x)*exp(x).
a(n) = n*A008599(n) = A195047(2*n). (End)

A195321 a(n) = 18*n^2.

Original entry on oeis.org

0, 18, 72, 162, 288, 450, 648, 882, 1152, 1458, 1800, 2178, 2592, 3042, 3528, 4050, 4608, 5202, 5832, 6498, 7200, 7938, 8712, 9522, 10368, 11250, 12168, 13122, 14112, 15138, 16200, 17298, 18432, 19602, 20808, 22050, 23328, 24642, 25992, 27378, 28800, 30258, 31752

Offset: 0

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 0, in the direction 0, 18, ..., in the square spiral whose vertices are the generalized hendecagonal numbers A195160. Semi-axis opposite to A195316 in the same spiral.
Area of a square with diagonal 6n. - Wesley Ivan Hurt, Jun 19 2014
Number of identical tessellation tiles that are composed of 48 equilateral edge joined triangles that can be formed into a order n hexagon. The example tiles shown in the link below are tessellated with eight sphinx tiles. See A291582. - Craig Knecht, Sep 02 2017

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

Bisection of A195147.
Cf. A016802, A033581, A033583, A135453, A139098, A144555, A195322, A195323.
Cf. A000290, A001105, A008600, A016766, A033428, A195160, A195316, A291582.

[18*n^2:n in [0..40]]; // Vincenzo Librandi, Sep 20 2011

A195321:=n->18*n^2; seq(A195321(n), n=0..50); # Wesley Ivan Hurt, Jun 19 2014

18 Range[0, 50]^2 (* or *) CoefficientList[Series[18 x*(1 + x)/(1 - x)^3, {x, 0, 30}], x] (* Wesley Ivan Hurt, Jun 20 2014 *)
LinearRecurrence[{3,-3,1},{0,18,72},50] (* Harvey P. Dale, Mar 26 2023 *)

a(n)=18*n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 18*A000290(n) = 9*A001105(n) = 6*A033428(n) = 3*A033581(n) = 2*A016766(n).
G.f.: 18*x*(1+x)/(1-x)^3. - Wesley Ivan Hurt, Jun 20 2014
From Elmo R. Oliveira, Dec 01 2024: (Start)
E.g.f.: 18*x*(1 + x)*exp(x).
a(n) = n*A008600(n) = A195147(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A195322 a(n) = 20*n^2.

Original entry on oeis.org

0, 20, 80, 180, 320, 500, 720, 980, 1280, 1620, 2000, 2420, 2880, 3380, 3920, 4500, 5120, 5780, 6480, 7220, 8000, 8820, 9680, 10580, 11520, 12500, 13520, 14580, 15680, 16820, 18000, 19220, 20480, 21780, 23120, 24500, 25920, 27380, 28880, 30420, 32000, 33620, 35280

Offset: 0

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 0, in the direction 0, 20, ..., in the square spiral whose vertices are the generalized dodecagonal numbers A195162. Semiaxis opposite to A195317 in the same spiral.
a(n) is the sum of all the integers less than 10*n which are not multiple of 2 or 5. a(2) = (1 + 3 + 7 + 9) + (11 + 13 + 17 + 19) = 20 + 60 = 80 = 20 * 2^2. (Link Crux Mathematicorum). - Bernard Schott, May 15 2017
Number of terms less than 10^k (k=0, 1, 2, ...): 1, 1, 3, 8, 23, 71, 224, 708, 2237, 7072, 22361, 70711, ... - Muniru A Asiru, Feb 01 2018

			From _Muniru A Asiru_, Feb 01 2018: (Start)
n=0, a(0) = 20*0^2 = 0.
n=1, a(1) = 20*1^2 = 20.
n=1, a(2) = 20*2^2 = 80.
n=1, a(3) = 20*3^2 = 180.
n=1, a(4) = 20*4^2 = 320.
...
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Léo Sauvé, Problem 53, Crux Mathematicorum, Vol. 1, Nov. 1975, page 88.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195148.
Cf. A016802, A033581, A033583, A135453, A139098, A144555, A195321, A195323.
Cf. A000290, A001105, A005899, A010014, A016742, A033429, A195162, A195317.
Cf. A008602.

List([0..10^3],n->20*n^2); # Muniru A Asiru, Feb 01 2018

[20*n^2: n in [0..40]]; // Vincenzo Librandi, Sep 20 2011

a := n -> 20*n^2; seq(a(n), n=0..10^3); # Muniru A Asiru, Feb 01 2018

20 Range[0, 40]^2 (* or *) LinearRecurrence[{3, -3, 1}, {0, 20, 80}, 50] (* Harvey P. Dale, Jan 18 2013 *)

a(n) = 20*n^2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 20*A000290(n) = 10*A001105(n) = 5*A016742(n) = 4*A033429(n) = 2*A033583(n).
a(0)=0, a(1)=20, a(2)=80; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jan 18 2013
a(n) = A010014(n) - A005899(n) for n > 0. - R. J. Cano, Sep 29 2015
From Elmo R. Oliveira, Nov 30 2024: (Start)
G.f.: 20*x*(1 + x)/(1-x)^3.
E.g.f.: 20*x*(1 + x)*exp(x).
a(n) = n*A008602(n) = A195148(2*n). (End)

A152742 13 times the squares: a(n) = 13*n^2.

Original entry on oeis.org

0, 13, 52, 117, 208, 325, 468, 637, 832, 1053, 1300, 1573, 1872, 2197, 2548, 2925, 3328, 3757, 4212, 4693, 5200, 5733, 6292, 6877, 7488, 8125, 8788, 9477, 10192, 10933, 11700, 12493, 13312, 14157, 15028, 15925, 16848, 17797, 18772

Offset: 0

Omar E. Pol, Dec 12 2008

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. A000290, A135453.

[13*n^2: n in [0..50]]; // G. C. Greubel, Sep 01 2018

13*Range[0,40]^2 (* or *) LinearRecurrence[{3,-3,1},{0,13,52},40] (* Harvey P. Dale, Feb 18 2015 *)

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

a(n) = 13*A000290(n).
a(n) = a(n-1) +26*n -13 (with a(0)=0). - Vincenzo Librandi, Nov 26 2010
a(0)=0, a(1)=13, a(2)=52, a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - Harvey P. Dale, Feb 18 2015
From G. C. Greubel, Sep 01 2018:(Start)
G.f.: 13*x*(1+x)/(1-x)^3.
E.g.f.: 13*(1+x)*exp(x). (End)
From Amiram Eldar, Feb 03 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/78.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/156.
Product_{n>=1} (1 + 1/a(n)) = sqrt(13)*sinh(Pi/sqrt(13))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(13)*sin(Pi/sqrt(13))/Pi. (End)

A195323 a(n) = 22*n^2.

Original entry on oeis.org

0, 22, 88, 198, 352, 550, 792, 1078, 1408, 1782, 2200, 2662, 3168, 3718, 4312, 4950, 5632, 6358, 7128, 7942, 8800, 9702, 10648, 11638, 12672, 13750, 14872, 16038, 17248, 18502, 19800, 21142, 22528, 23958, 25432, 26950, 28512, 30118, 31768, 33462, 35200, 36982, 38808

Offset: 0

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 0, in the direction 0, 22, ..., in the square spiral whose vertices are the generalized tridecagonal numbers A195313. Semi-axis opposite to A195318 in the same spiral.

Surface area of a rectangular prism with dimensions n, 2n and 3n. - Wesley Ivan Hurt, Apr 10 2015

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

Bisection of A195149.
Cf. A016802, A033581, A033583, A135453, A139098, A144555, A195321, A195322.
Cf. A000290, A001105, A008604, A033584, A195313, A195318.

[22*n^2 : n in [0..40]]; // Vincenzo Librandi, Sep 20 2011

A195323:=n->22*n^2: seq(A195323(n), n=0..80); # Wesley Ivan Hurt, Apr 10 2015

22Range[0,50]^2 (* or *) LinearRecurrence[{3,-3,1},{0,22,88},50] (* Harvey P. Dale, Sep 19 2011 *)

vector(50,n,22*(n-1)^2) \\ Derek Orr, Apr 10 2015

a(n) = 22*A000290(n) = 11*A001105(n) = 2*A033584(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Sep 19 2011
G.f.: 22*x*(1+x)/(1-x)^3. - Wesley Ivan Hurt, Apr 10 2015
From Elmo R. Oliveira, Dec 01 2024: (Start)
E.g.f.: 22*x*(1 + x)*exp(x).
a(n) = n*A008604(n) = A195149(2*n). (End)

A016910 a(n) = (6*n)^2.

Original entry on oeis.org

0, 36, 144, 324, 576, 900, 1296, 1764, 2304, 2916, 3600, 4356, 5184, 6084, 7056, 8100, 9216, 10404, 11664, 12996, 14400, 15876, 17424, 19044, 20736, 22500, 24336, 26244, 28224, 30276, 32400, 34596, 36864, 39204, 41616, 44100, 46656, 49284, 51984, 54756, 57600, 60516, 63504, 66564, 69696, 72900

Offset: 0

N. J. A. Sloane

Areas A of two classes of triangles with integer sides (a,b,c) where a = 9k, b=10k and c = 17k, or a = 3k, b = 25k and c = 26k for k=0,1,2,... These areas are given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) = (6k)^2, with the semiperimeter s = (a+b+c)/2. This sequence is a subsequence of A188158. - Michel Lagneau, Oct 11 2013

Sequence found by reading the line from 0, in the direction 0, 36, ..., in the square spiral whose vertices are the generalized 20-gonal numbers A218864. - Omar E. Pol, May 13 2018.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A089491, A188158, A243941.

Cf. similar sequences of the type k*n^2: A000290 (k=1), A001105 (k=2), A033428 (k=3), A016742 (k=4), A033429 (k=5), A033581 (k=6), A033582 (k=7), A139098 (k=8), A016766 (k=9), A033583 (k=10), A033584 (k=11), A135453 (k=12), A152742 (k=13), A144555 (k=14), A064761 (k=15), A016802 (k=16), A244630 (k=17), A195321 (k=18), A244631 (k=19), A195322 (k=20), A064762 (k=21), A195323 (k=22), A244632 (k=23), A195824 (k=24), A016850 (k=25), A244633 (k=26), A244634 (k=27), A064763 (k=28), A244635 (k=29), A244636 (k=30).

[(6*n)^2: n in [0..40]]; // Vincenzo Librandi, May 03 2011

(6*Range[0,40])^2 (* or *) LinearRecurrence[{3,-3,1},{0,36,144},40] (* Harvey P. Dale, Dec 25 2017 *)
Table[36 n^2, {n, 0, 45}] (* Omar E. Pol, Jun 07 2018 *)

a(n)=36*n^2 \\ Charles R Greathouse IV, Jun 10 2016

From Ilya Gutkovskiy, Jun 09 2016: (Start)
O.g.f.: 36*x*(1 + x)/(1 - x)^3.
E.g.f.: 36*x*(1 + x)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
Sum_{n>=1} 1/a(n) = Pi^2/216 = A086726. (End)
Product_{n>=1} a(n)/A136017(n) = Pi/3. - Fred Daniel Kline, Jun 09 2016
a(n) = t(9*n) - 9*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(9*n) - 9*A000217(n). - Bruno Berselli, Aug 31 2017
a(n) = 36*A000290(n) = 18*A001105(n) = 12*A033428 = 9*A016742(n) = 6*A033581(n) = 4*A016766(n) = 3*A135453(n) = 2*A195321(n). - Omar E. Pol, Jun 07 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/432. - Amiram Eldar, Jun 27 2020
From Amiram Eldar, Jan 25 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/6)/(Pi/6).
Product_{n>=1} (1 - 1/a(n)) = sin(Pi/6)/(Pi/6) = 3/Pi (A089491). (End)

A158463 a(n) = 12*n^2 - 1.

Original entry on oeis.org

-1, 11, 47, 107, 191, 299, 431, 587, 767, 971, 1199, 1451, 1727, 2027, 2351, 2699, 3071, 3467, 3887, 4331, 4799, 5291, 5807, 6347, 6911, 7499, 8111, 8747, 9407, 10091, 10799, 11531, 12287, 13067, 13871, 14699, 15551, 16427, 17327, 18251, 19199, 20171, 21167, 22187

Offset: 0

Vincenzo Librandi, Mar 19 2009

Related to Legendre polynomials - see Mma line. - N. J. A. Sloane, Nov 17 2009
One notices that this sequence produces an inordinate number of semiprimes, perhaps better than mere chance for large values of n. - J. M. Bergot, Jun 30 2011
Sequence found by reading the line from -1, in the direction -1, 11, ..., in the square spiral whose vertices are -1 together with the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012

			G.f. = -1 + 11*x + 47*x^2 + 107*x^3 + 191*x^4 + 299*x^5 + 431*x^6 + 587*x^7 + 767*x^8 + ...

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

Cf. A001082, A080663, A135453.

[12*n^2 - 1: n in [0..100]]; // G. C. Greubel, Sep 25 2018

Table[Numerator[LegendreP[2,2n]],{n,0,50}] (* N. J. A. Sloane, Nov 17 2009 *)
a[ n_] := 2 LegendreP[ 2, 2 n] (* Michael Somos, Jul 01 2011 *)
12*Range[0,40]^2-1 (* or *) LinearRecurrence[{3,-3,1},{-1,11,47},50] (* Harvey P. Dale, Jun 22 2019 *)

{a(n) = 12 * n^2 - 1} /* Michael Somos, Nov 12 2011 */

a(-n) = a(n).
G.f.: (1-14*x-11*x^2)/(x-1)^3. - R. J. Mathar, Aug 27 2011
a(n) = A135453(n) - 1. - Omar E. Pol, Jul 18 2012
From Amiram Eldar, Feb 04 2021: (Start)
Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(12))*cot(Pi/sqrt(12)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(12))*csc(Pi/sqrt(12)) - 1)/2.
Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(12))*csc(Pi/sqrt(12)).
Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(12))*sin(Pi/sqrt(6))/sqrt(2). (End)
From Gerry Martens, Apr 06 2024: (Start)
a(n) = Re((2*n*i-1)^3).
a(n) = -8*(1/4+n^2)^(3/2)*cos(3*arctan(2*n)). (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: exp(x)*(12*x^2 + 12*x - 1).
a(n) = A080663(2*n) for n > 0. (End)

Minor edits by N. J. A. Sloane, Nov 17 2009

Comment rewritten by Bruno Berselli, Aug 27 2011

A195824 a(n) = 24*n^2.

Original entry on oeis.org

0, 24, 96, 216, 384, 600, 864, 1176, 1536, 1944, 2400, 2904, 3456, 4056, 4704, 5400, 6144, 6936, 7776, 8664, 9600, 10584, 11616, 12696, 13824, 15000, 16224, 17496, 18816, 20184, 21600, 23064, 24576, 26136, 27744, 29400, 31104, 32856, 34656, 36504, 38400, 40344

Offset: 0

Omar E. Pol, Sep 28 2011

Sequence found by reading the line from 0, in the direction 0, 24, ..., in the square spiral whose vertices are the generalized tetradecagonal numbers A195818.

Surface area of a cube with side 2n. - Wesley Ivan Hurt, Aug 05 2014

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

Bisection of A195158.

Cf. A000290, A001105, A008606, A016742, A033428, A033581, A135453, A139098, A195818.

[24*n^2 : n in [0..50]]; // Wesley Ivan Hurt, Aug 05 2014

I:=[0,24,96]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Aug 06 2014

A195824:=n->24*n^2: seq(A195824(n), n=0..50); # Wesley Ivan Hurt, Aug 05 2014

24 Range[0, 30]^2 (* or *) Table[24 n^2, {n, 0, 30}] (* or *) CoefficientList[Series[24 x (1 + x)/(1 - x)^3, {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 05 2014 *)
LinearRecurrence[{3,-3,1},{0,24,96},40] (* Harvey P. Dale, Nov 11 2017 *)

a(n) = 24*n^2; \\ Michel Marcus, Aug 05 2014

a(n) = 24*A000290(n) = 12*A001105(n) = 8*A033428(n) = 6*A016742(n) = 4*A033581(n) = 3*A139098(n) = 2*A135453(n).
From Wesley Ivan Hurt, Aug 05 2014: (Start)
G.f.: 24*x*(1+x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Elmo R. Oliveira, Dec 01 2024: (Start)
E.g.f.: 24*x*(1 + x)*exp(x).
a(n) = n*A008606(n) = A195158(2*n). (End)

cp's OEIS Frontend

A069190 Centered 24-gonal numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

Formula

A244630 a(n) = 17*n^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Scala

Formula

A195321 a(n) = 18*n^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A195322 a(n) = 20*n^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

A152742 13 times the squares: a(n) = 13*n^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula