A049598 -id:A049598 - cp's OEIS frontend

A152741 13 times triangular numbers.

0, 13, 39, 78, 130, 195, 273, 364, 468, 585, 715, 858, 1014, 1183, 1365, 1560, 1768, 1989, 2223, 2470, 2730, 3003, 3289, 3588, 3900, 4225, 4563, 4914, 5278, 5655, 6045, 6448, 6864, 7293, 7735, 8190, 8658, 9139, 9633, 10140, 10660, 11193, 11739, 12298, 12870

Offset: 0

Omar E. Pol, Dec 12 2008

Sequence found by reading the line from 0, in the direction 0, 13,... and the same line from 0, in the direction 0, 39,..., in the square spiral whose vertices are the generalized 15-gonal numbers. - Omar E. Pol, Oct 03 2011

Sum of the numbers from 6n to 7n. - Wesley Ivan Hurt, Dec 22 2015

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. A000217, A049598, A069126.

[13*n*(n+1)/2 : n in [0..60]]; // Wesley Ivan Hurt, Dec 22 2015

A152741:=n->13*n*(n+1)/2: seq(A152741(n), n=0..60); # Wesley Ivan Hurt, Dec 22 2015

Table[13*n*(n-1)/2, {n,100}] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2011 *)
CoefficientList[Series[13 x/(1 - x)^3, {x, 0, 50}], x] (* Wesley Ivan Hurt, Dec 22 2015 *)

a(n)=13*n*(n+1)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 13*n*(n+1)/2 = 13 * A000217(n).
a(n) = a(n-1)+13*n (with a(0)=0). - Vincenzo Librandi, Nov 26 2010
a(n) = A069126(n+1) - 1. - Omar E. Pol, Oct 03 2011
From Wesley Ivan Hurt, Dec 22 2015: (Start)
G.f.: 13*x/(1-x)^3.
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>2.
a(n) = Sum_{i=6n..7n} i. (End)
E.g.f.: 13*x*(2+x)*exp(x)/2. - G. C. Greubel, Sep 01 2018
From Amiram Eldar, Feb 21 2023: (Start)
Sum_{n>=1} 1/a(n) = 2/13.
Sum_{n>=1} (-1)^(n+1)/a(n) = (4*log(2) - 2)/13.
Product_{n>=1} (1 - 1/a(n)) = -(13/(2*Pi))*cos(sqrt(21/13)*Pi/2).
Product_{n>=1} (1 + 1/a(n)) = (13/(2*Pi))*cos(sqrt(5/13)*Pi/2). (End)

A195158 Concentric 24-gonal numbers.

Original entry on oeis.org

0, 1, 24, 49, 96, 145, 216, 289, 384, 481, 600, 721, 864, 1009, 1176, 1345, 1536, 1729, 1944, 2161, 2400, 2641, 2904, 3169, 3456, 3745, 4056, 4369, 4704, 5041, 5400, 5761, 6144, 6529, 6936, 7345, 7776, 8209, 8664, 9121, 9600, 10081, 10584, 11089

Offset: 0

Omar E. Pol, Sep 28 2011

Sequence found by reading the line from 0, in the direction 0, 24, ..., and the same line from 1, in the direction 1, 49, ..., in the square spiral whose vertices are the generalized tetradecagonal numbers A195818. Main axis, perpendicular to A049598 in the same spiral.

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

Column 24 of A195040.

Cf. A032527, A032528, A195143, A195149, A195058.

[(12*n^2+5*(-1)^n-5)/2: n in [0..50]]; // Vincenzo Librandi, Sep 30 2011

LinearRecurrence[{2,0,-2,1},{0,1,24,49},50] (* Harvey P. Dale, Jan 28 2021 *)

a(n)=6*n^2+5*((-1)^n-1)/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 6*n^2 + 5*((-1)^n-1)/2.
a(n) = -a(n-1) + A069190(n). - Vincenzo Librandi, Sep 30 2011
From Colin Barker, Sep 16 2012: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: x*(1+22*x+x^2)/((1-x)^3*(1+x)). (End)
Sum_{n>=1} 1/a(n) = Pi^2/144 + tan(sqrt(5/6)*Pi/2)*Pi/(4*sqrt(30)). - Amiram Eldar, Jan 17 2023

A263226 a(n) = 15n^2 - 13n.

Original entry on oeis.org

0, 2, 34, 96, 188, 310, 462, 644, 856, 1098, 1370, 1672, 2004, 2366, 2758, 3180, 3632, 4114, 4626, 5168, 5740, 6342, 6974, 7636, 8328, 9050, 9802, 10584, 11396, 12238, 13110, 14012, 14944, 15906, 16898, 17920, 18972, 20054, 21166, 22308, 23480, 24682, 25914

Offset: 0

Emeric Deutsch, Oct 12 2015

For n>=3, a(n) = the Wiener index of the Jahangir graph J_{3,n}. The Jahangir graph J_{3,n} is a connected graph consisting of a cycle graph C(3n) and one additional center vertex that is adjacent to n vertices of C(3n) at distances 3 to each other on C(3n).

The Hosoya polynomial of J_(3,n) is 4nx + (1/2)n(n+9)x^2 + 2n(n-1)x^3 + n(2n-5)x^4.

M. R. Farahani, The Wiener index and Hosoya polynomial of a class of Jahangir graphs J_{3,m}, Fundamental J. Math. and Math. Sci., 3 (1), 91-96, 2015.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A049598, A263227, A263228, A263229, A263231.

[15*n^2-13*n: n in [0..50]]; // Bruno Berselli, Oct 15 2015

Maple
```
seq(15*n^2-13*n, n = 0 .. 40);
```

Table[15 n^2 - 13 n, {n, 0, 40}] (* Vincenzo Librandi, Oct 13 2015 *)
LinearRecurrence[{3,-3,1},{0,2,34},50] (* Harvey P. Dale, Jul 27 2018 *)

vector(50, n, n--; 15*n^2 - 13*n) \\ Altug Alkan, Oct 12 2015

G.f.: 2*x*(1 + 14*x)/(1 - x)^3. - Vincenzo Librandi, Oct 13 2015

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Oct 13 2015

A263227 a(n) = n(67n - 89)/2.

Original entry on oeis.org

0, -11, 45, 168, 358, 615, 939, 1330, 1788, 2313, 2905, 3564, 4290, 5083, 5943, 6870, 7864, 8925, 10053, 11248, 12510, 13839, 15235, 16698, 18228, 19825, 21489, 23220, 25018, 26883, 28815, 30814, 32880, 35013, 37213, 39480, 41814, 44215, 46683, 49218, 51820

Offset: 0

Emeric Deutsch, Oct 12 2015

For n>=3, a(n) = the hyper-Wiener index of the Jahangir graph J_{3,n}. The Jahangir graph J_{3,n} is a connected graph consisting of a cycle graph C(3n) and one additional center vertex that is adjacent to n vertices of C(3n) at distances 3 to each other on C(3n).

The Hosoya polynomial of J_(3,n) is 4nx + (1/2)n(n+9)x^2 + 2n(n-1)x^3 + n(2n-5)x^4.

M. R. Farahani, The Wiener index and Hosoya polynomial of a class of Jahangir graphs J_{3,m}, Fundamental J. Math. and Math. Sci., 3 (1), 91-96, 2015.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A049598, A263226, A263228, A263229, A263231.

[n*(67*n-89)/2: n in [0..40]]; // Bruno Berselli, Oct 15 2015

Maple
```
seq((1/2)*n*(67*n-89), n = 0 .. 40);
```

Table[n (67 n - 89)/2, {n, 0, 40}] (* Vincenzo Librandi, Oct 13 2015 *)

vector(50, n, n--; n*(67*n-89)/2) \\ Altug Alkan, Oct 12 2015

G.f.: x*(-11+78*x)/(1-x)^3. - Vincenzo Librandi, Oct 13 2015

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Oct 13 2015

A263228 a(n) = 2n(16*n - 13).

Original entry on oeis.org

0, 6, 76, 210, 408, 670, 996, 1386, 1840, 2358, 2940, 3586, 4296, 5070, 5908, 6810, 7776, 8806, 9900, 11058, 12280, 13566, 14916, 16330, 17808, 19350, 20956, 22626, 24360, 26158, 28020, 29946, 31936, 33990, 36108, 38290, 40536, 42846, 45220, 47658, 50160

Offset: 0

Emeric Deutsch, Oct 13 2015

For n>=3, a(n) = the Wiener index of the Jahangir graph J_{4,n}. The Jahangir graph J_{4,n} is a connected graph consisting of a cycle graph C(4*n) and one additional center vertex that is adjacent to n vertices of C(4*n) at distances 4 to each other on C(4*n). In the Farahani reference the expression in Theorem 2 is accidentally incorrect; it should be 2*m*(16*m - 13).

The Hosoya polynomial of J_{4,n} is 5*n*x + n*(n+11)*x^2/2 + n*(2*n+1)*x^3 + n*(3*n-4)*x^4 + 2*n*(n-2)*x^5 + n*(n-3)*x^6/2 (see the Farahani reference, p. 234, last line; however, the expression in Theorem 1, p. 233, is accidentally incorrect).

M. R. Farahani, Hosoya polynomial and of Jahangir graphs J_{4,m}, Global J. Math, 3 (1), 232-236, 2015.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A049598, A263226, A263227, A263229, A263231.

[2*n*(16*n-13): n in [0..60]]; // Vincenzo Librandi, Oct 15 2015

Maple
```
seq(32*n^2 - 26*n, n=0..40);
```

Table[2 n (16 n - 13), {n, 0, 40}] (* Bruno Berselli, Oct 15 2015 *)

vector(50, n, n--; 2*n*(16*n-13)) \\ Altug Alkan, Oct 15 2015

G.f.: 2*x*(3+29*x)/(1-x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

A263229 a(n) = 4n(21*n - 26).

Original entry on oeis.org

0, -20, 128, 444, 928, 1580, 2400, 3388, 4544, 5868, 7360, 9020, 10848, 12844, 15008, 17340, 19840, 22508, 25344, 28348, 31520, 34860, 38368, 42044, 45888, 49900, 54080, 58428, 62944, 67628, 72480, 77500, 82688, 88044, 93568, 99260, 105120, 111148, 117344, 123708, 130240

Offset: 0

Emeric Deutsch, Oct 13 2015

For n>=3, a(n) = the hyper-Wiener index of the Jahangir graph J_{4,n}.

See A263228 for more comments.

M. R. Farahani, Hosoya polynomial and of Jahangir graphs J_{4,m}, Global J. Math, 3 (1), 232-236, 2015.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A049598, A263226, A263227, A263228, A263231.

[4*n*(21*n-26): n in [0..20]]; // Vincenzo Librandi, Oct 15 2015

Maple
```
seq(84*n^2 - 104*n, n=0..40);
```

Table[4 n (21 n - 26), {n, 0, 40}] (* Bruno Berselli, Oct 15 2015 *)

vector(50, n, n--; 4*n*(21*n-26)) \\ Altug Alkan, Oct 15 2015

G.f.: 4*x*(47*x-5)/(1-x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

A221912 Partial sums of floor(n/12).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 125, 130, 135, 140, 145, 150, 155

Offset: 0

Philippe Deléham, Mar 27 2013

Apart from the initial zeros, the same as A008730.

			..0....0....0....0....0....0....0....0....0....0....0....0
..1....2....3....4....5....6....7....8....9...10...11...12
.14...16...18...20...22...24...26...28...30...32...34...36
.39...42...45...48...51...54...57...60...63...66...69...72
.76...80...84...88...92...96..100..104..108..112..116..120
125..130..135..140..145..150..155..160..165..170..175..180
186..192..198..204..210..216..222..228..234..240..246..252
259..266..273..280..287..294..301..308..315..322..329..336
344..352..360..368..376..384..392..400..408..416..424..432
441..450..459..468..477..486..495..504..513..522..531..540
...

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,0,0,1,-2,1).

Cf. A000217, A002620, A130518, A130519, A130520, A174709, A174738, A118729, A218470, A131242, A218530.

Accumulate[Floor[Range[0,70]/12]] (* or *) LinearRecurrence[{2,-1,0,0,0,0,0,0,0,0,0,1,-2,1},{0,0,0,0,0,0,0,0,0,0,0,0,1,2},70] (* Harvey P. Dale, Mar 23 2015 *)

a(12n) = A051866(n).
a(12n+1) = A139267(n).
a(12n+2) = A094159(n).
a(12n+3) = A033579(n).
a(12n+4) = A049452(n).
a(12n+5) = A033581(n).
a(12n+6) = A049453(n).
a(12n+7) = A033580(n).
a(12n+8) = A195319(n).
a(12n+9) = A202804(n).
a(12n+10) = A211014(n).
a(12n+11) = A049598(n).
G.f.: x^12/((1-x)^2*(1-x^12)).
a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=0, a(5)=0, a(6)=0, a(7)=0, a(8)=0, a(9)=0, a(10)=0, a(11)=0, a(12)=1, a(13)=2, a(n)=2*a(n-1)- a(n-2)+ a(n-12)- 2*a(n-13)+ a(n-14). - Harvey P. Dale, Mar 23 2015

A008730 Molien series 1/((1-x)^2(1-x^12)) for 3-dimensional group [2,n] = 22n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 186, 192, 198, 204

Offset: 0

N. J. A. Sloane

			..1....2....3....4....5....6....7....8....9...10...11...12
.14...16...18...20...22...24...26...28...30...32...34...36
.39...42...45...48...51...54...57...60...63...66...69...72
.76...80...84...88...92...96..100..104..108..112..116..120
125..130..135..140..145..150..155..160..165..170..175..180
186..192..198..204..210..216..222..228..234..240..246..252
259..266..273..280..287..294..301..308..315..322..329..336
344..352..360..368..376..384..392..400..408..416..424..432
441..450..459..468..477..486..495..504..513..522..531..540
550..560..570..580..590..600..610..620..630..640..650..660
...
The columns are: A051866, A139267, A094159, A033579, A049452, A033581, A049453, A033580, A195319, A202804, A211014, A049598
- _Philippe Deléham_, Apr 03 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 195
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1).

Cf. A001840, A001972, A008724, A008725, A008726, A008727, A008728, A008729, A008732. - Vladimir Joseph Stephan Orlovsky, Mar 14 2010

Cf. A221912

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)^2*(1-x^12)) )); // G. C. Greubel, Jul 30 2019

seq(coeff(series(1/(1-x)^2/(1-x^12), x, n+1), x, n), n=0..80);

CoefficientList[Series[1/((1-x)^2*(1-x^12)), {x,0,70}], x] (* Vincenzo Librandi, Jun 11 2013 *)
LinearRecurrence[{2,-1,0,0,0,0,0,0,0,0,0,1,-2,1},{1,2,3,4,5,6,7,8,9,10,11,12,14,16},70] (* Harvey P. Dale, Jan 01 2024 *)

my(x='x+O('x^70)); Vec(1/((1-x)^2*(1-x^12))) \\ G. C. Greubel, Jul 30 2019

(1/((1-x)^2*(1-x^12))).series(x, 70).coefficients(x, sparse=False) # G. C. Greubel, Jul 30 2019

G.f. 1/( (1-x)^3 * (1+x) *(1+x+x^2) *(1-x+x^2) * (1+x^2) *(1-x^2+x^4)). - R. J. Mathar, Aug 11 2021
From Mitch Harris, Sep 08 2008: (Start)
a(n) = Sum_{j=0..n+12} floor(j/12).
a(n-12) = (1/2)*floor(n/12)*(2*n - 10 - 12*floor(n/12)). (End)
a(n) = A221912(n+12). - Philippe Deléham, Apr 03 2013

More terms from Vladimir Joseph Stephan Orlovsky, Mar 14 2010

A060834 a(n) = 6n^2 + 6n + 31.

Original entry on oeis.org

31, 43, 67, 103, 151, 211, 283, 367, 463, 571, 691, 823, 967, 1123, 1291, 1471, 1663, 1867, 2083, 2311, 2551, 2803, 3067, 3343, 3631, 3931, 4243, 4567, 4903, 5251, 5611, 5983, 6367, 6763, 7171, 7591, 8023, 8467, 8923, 9391, 9871, 10363, 10867, 11383

Offset: 0

Jason Earls, May 02 2001

First 29 values are primes.
From Peter Bala, Apr 18 2018: (Start)
Let P(n) = 6*n^2 + 6*n + 31. The polynomial P(2*n-14) = 24*n^2 - 660*n + 4567 takes distinct prime values for n = 0 to 28.
The value of the polynomial 2*P(3/2*(n-10)) = 27*n^2 - 522*n + 2582 for n = 0 to 22 is either double a prime or a prime (alternately).
The value of the polynomial 4*P(4/3*(n-9)) = 32*n^2 - 552*n + 2469 for n = 0 to 28 is either prime or 3 times a prime, except when n = 16. (End)
Also, numbers k such that 2*k/3 - 2/3 - 19 is a perfect square. - Bruno Berselli, Apr 23 2018
Equivalently, numbers k such that 6*k - 177 is a square. - Vincenzo Librandi, Apr 23 2018

			a(29)=4903, prime. a(30)=5251, nonprime.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 145.
Donald D. Spencer, Computers in Number Theory, Computer Science Press, Rockville, MD, 1982, pp. 118-119.

Harry J. Smith, Table of n, a(n) for n = 0..1000
T. Piezas, A collection of algebraic identities. 0023: Part 2, Prime Generating Polynomials, Section IV.
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A049598, A060844, A005846.

List([0..80],n->6*n^2+6*n+31); # Muniru A Asiru, Apr 22 2018

Table[6n^2+6n+31,{n,0,60}] (* or *) LinearRecurrence[{3,-3,1},{31,43,67},60] (* Harvey P. Dale, Aug 09 2011 *)

a(n) = { 6*n^2 + 6*n + 31 } \\ Harry J. Smith, Jul 19 2009

From R. J. Mathar, Feb 05 2008: (Start)
O.g.f.: -(31-50*x+31*x^2)/(-1+x)^3.
a(n) = A049598(n)+31. (End)
a(0)=31, a(1)=43, a(2)=67, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Aug 09 2011
E.g.f.: exp(x)*(31 + 12*x + 6*x^2). - Stefano Spezia, Dec 26 2024

More terms from Larry Reeves (larryr(AT)acm.org), May 07 2001

A272975 Numbers that are congruent to {0,7} mod 12.

Original entry on oeis.org

0, 7, 12, 19, 24, 31, 36, 43, 48, 55, 60, 67, 72, 79, 84, 91, 96, 103, 108, 115, 120, 127, 132, 139, 144, 151, 156, 163, 168, 175, 180, 187, 192, 199, 204, 211, 216, 223, 228, 235, 240, 247, 252, 259, 264, 271, 276, 283, 288, 295, 300, 307, 312, 319, 324

Offset: 1

Wesley Ivan Hurt, May 30 2016

Numbers that are not congruent to {1, 2, 3, 4, 5, 6, 8, 9, 10, 11} mod 12.

Bisection of A083032.

David Lovler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A008594, A017605, A083032, A049453, A049598.

[n : n in [0..400] | n mod 12 in [0, 7]];

A272975:=n->(12*n-11+(-1)^n)/2: seq(A272975(n), n=1..100);

Mathematica
```
Table[(12n - 11 + (-1)^n)/2, {n, 80}]
```

concat(0, Vec(x^2*(7+5*x)/((x-1)^2*(x+1)) + O(x^99))) \\ Altug Alkan, May 31 2016

G.f.: x^2*(7+5*x) / ((x-1)^2*(x+1)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3.
a(n) = (12*n - 11 + (-1)^n)/2.
a(2k) = A017605(k-1) k>0, a(2k-1) = A008594(k-1) k>0, a(2k)-a(2k-1) = 7.
a(n)-a(-n) = A008594(n) for n>0.
Sum_{i=1..n} a(2*i) = A049453(n) for n>0.
Sum_{i=1..n} a(2*i-1) = A049598(n-1) for n>0.
E.g.f.: 5 + ((12*x - 11)*exp(x) + exp(-x))/2. - David Lovler, Sep 04 2022
Sum_{n>=2} (-1)^n/a(n) = log(2)/4 + log(3)/8 - ((sqrt(3)-1)*Pi + 2*(sqrt(3)+3)*log(sqrt(3)+2))/(24*(sqrt(3)+1)). - Amiram Eldar, Sep 17 2023

cp's OEIS Frontend

A152741 13 times triangular numbers.

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A195158 Concentric 24-gonal numbers.

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A263226 a(n) = 15*n^2 - 13*n.

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A263227 a(n) = n*(67*n - 89)/2.

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A263228 a(n) = 2*n*(16*n - 13).

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A263229 a(n) = 4*n*(21*n - 26).

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A221912 Partial sums of floor(n/12).

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A263226 a(n) = 15n^2 - 13n.

A263227 a(n) = n(67n - 89)/2.

A263228 a(n) = 2n(16*n - 13).

A263229 a(n) = 4n(21*n - 26).

A008730 Molien series 1/((1-x)^2(1-x^12)) for 3-dimensional group [2,n] = 22n.

A060834 a(n) = 6n^2 + 6n + 31.