A226492 -id:A226492 - cp's OEIS frontend

A094159 3 times hexagonal numbers: a(n) = 3n(2*n-1).

0, 3, 18, 45, 84, 135, 198, 273, 360, 459, 570, 693, 828, 975, 1134, 1305, 1488, 1683, 1890, 2109, 2340, 2583, 2838, 3105, 3384, 3675, 3978, 4293, 4620, 4959, 5310, 5673, 6048, 6435, 6834, 7245, 7668, 8103, 8550, 9009, 9480, 9963, 10458, 10965, 11484

Offset: 0

N. J. A. Sloane, May 05 2004

Column 3 of A048790.
Sequence found by reading the line from 0, in the direction 0, 3, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011
a(n) is the sum of all perimeters of triangles having two sides of length n. For n=4 one has seven triangles with two sides of length 4 and the other of lengths 1..7. - J. M. Bergot, Mar 26 2014
a(n) is the Wiener index of the complete tripartite graph K_{n,n,n}. - Eric W. Weisstein, Sep 07 2017
Sequence found by reading the line from 0, in the direction 0, 3, ..., in a spiral on an equilateral triangular lattice. - Hans G. Oberlack, Dec 08 2018

Dan Hoey, Bill Gosper and Richard C. Schroeppel, Discussions in Math-Fun Mailing list, circa Jul 13 1999.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hans G. Oberlack, Triangle spiral.
R. C. Schroeppel, A few mathematical experiments, Experimental Mathematics Workshop, Oakland, California, March 30, 2004.
Eric Weisstein's World of Mathematics, Complete Tripartite Graph.
Eric Weisstein's World of Mathematics, Wiener Index.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000384, A048790.
3 times n-gonal numbers: A045943, A033428, A062741, A152773, A152751, A152759, A152767, A153783, A153448, A153875.
Essentially a bisection of A045943. - Omar E. Pol, Sep 17 2011
Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=12: see Comments lines of A226492.

List([0..50], n -> 3*n*(2*n-1)); # G. C. Greubel, Dec 07 2018

[3*n*(2*n-1): n in [0..50]]; // G. C. Greubel, Dec 07 2018

A094159:=n->3*n*(2*n-1); seq(A094159(n), n=0..40); # Wesley Ivan Hurt, Mar 28 2014

CoefficientList[Series[3x(1+3x)/(1-x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 19 2013 *)
Table[3n(2n-1), {n, 0, 50}] (* or *) 3*PolygonalNumber[6, Range[0, 50]] (* or *) LinearRecurrence[{3, -3, 1}, {3, 18, 45}, {0, 50}] (* Eric W. Weisstein, Sep 07 2017 *)

a(n)=3*n*(2*n-1) \\ Charles R Greathouse IV, Sep 24 2015

[3*n*(2*n-1) for n in range(50)] # G. C. Greubel, Dec 07 2018

a(n) = 6*n^2 - 3*n = 3*n*(2*n-1) = 3*A000384(n). - Omar E. Pol, Dec 11 2008
a(n) = 12*n + a(n-1) - 9 with n > 0, a(0)=0. - Vincenzo Librandi, Nov 16 2010
G.f.: 3*x*(1+3*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
Sum_{n>0} 1/a(n) = (2/3)*log(2). - Enrique Pérez Herrero, Jun 04 2015
E.g.f.: 3*x*(1+2*x)*exp(x). - G. C. Greubel, Dec 07 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/6 - log(2)/3. - Amiram Eldar, Jan 10 2022

More terms from Vladimir Joseph Stephan Orlovsky, Nov 16 2008

Definition improved, offset corrected and edited by Omar E. Pol, Dec 11 2008

A152751 3 times octagonal numbers: a(n) = 3n(3*n-2).

Original entry on oeis.org

0, 3, 24, 63, 120, 195, 288, 399, 528, 675, 840, 1023, 1224, 1443, 1680, 1935, 2208, 2499, 2808, 3135, 3480, 3843, 4224, 4623, 5040, 5475, 5928, 6399, 6888, 7395, 7920, 8463, 9024, 9603, 10200, 10815, 11448, 12099, 12768, 13455, 14160, 14883, 15624, 16383, 17160

Offset: 0

Omar E. Pol, Dec 12 2008

a(n) also can be represented as n concentric triangles (see example). - Omar E. Pol, Aug 21 2011

			From _Omar E. Pol_, Aug 21 2011: (Start)
Illustration of initial terms as concentric triangles:
.
.                                          o
.                                         o o
.                                        o   o
.                                       o     o
.                 o                    o   o   o
.                o o                  o   o o   o
.               o   o                o   o   o   o
.              o     o              o   o     o   o
.    o        o   o   o            o   o   o   o   o
.   o o      o   o o   o          o   o   o o   o   o
.           o           o        o   o           o   o
.          o o o o o o o o      o   o o o o o o o o   o
.                              o                       o
.                             o o o o o o o o o o o o o o
.
.    3            24                       63
(End)

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

Cf. A000567, A064201, A139267, A152995.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152759, A152767, A153783, A153448, A153875.
Cf. A033581, A085250, A152734, A194273. - Omar E. Pol, Aug 21 2011
Cf. numbers of the form n*(n*k - k + 6)/2, this sequence is the case k=18: see Comments lines of A226492.

s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,3,6!,18}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2009 *)
3*PolygonalNumber[8,Range[0,40]] (* Harvey P. Dale, May 08 2022 *)

a(n)=3*n*(3*n-2) \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 9*n^2 - 6*n = 3*A000567(n) = A064201(n)/3.
a(n) = a(n-1) + 18*n - 15 with n > 0, a(0)=0. - Vincenzo Librandi, Nov 26 2010
G.f.: 3*x*(1+5*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
From Elmo R. Oliveira, Dec 25 2024: (Start)
E.g.f.: 3*exp(x)*x*(1 + 3*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3.
a(n) = n + A152995(n). (End)

A152773 3 times heptagonal numbers: a(n) = 3n(5*n-3)/2.

Original entry on oeis.org

0, 3, 21, 54, 102, 165, 243, 336, 444, 567, 705, 858, 1026, 1209, 1407, 1620, 1848, 2091, 2349, 2622, 2910, 3213, 3531, 3864, 4212, 4575, 4953, 5346, 5754, 6177, 6615, 7068, 7536, 8019, 8517, 9030, 9558, 10101, 10659, 11232, 11820, 12423, 13041, 13674, 14322, 14985

Offset: 0

Omar E. Pol, Dec 13 2008

Also the number of 6-cycles in the (n+5)-triangular honeycomb acute knight graph. - Eric W. Weisstein, Jun 25 2017

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Graph Cycle.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000566, A001622, A135706, A226489.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152751, A152759, A152767, A153783, A153448, A153875.
Cf. numbers of the form n*(n*k - k + 6)/2, this sequence is the case k=15: see Comments lines of A226492.
Cf. A002378 (3-cycles in triangular honeycomb acute knight graph), A045943 (4-cycles), A028896 (5-cycles).

Table[3 n (5 n - 3)/2, {n, 0, 50}] (* Harvey P. Dale, May 08 2012 *)
LinearRecurrence[{3, -3, 1}, {0, 3, 21}, 50] (* Harvey P. Dale, May 08 2012 *)
CoefficientList[Series[-((3 x^5 (1 + 4 x))/(-1 + x)^3), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 25 2017 *)

a(n)=3*n*(5*n-3)/2 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = (15*n^2 - 9*n)/2 = 3*A000566(n).
a(n) = a(n-1) + 15*n - 12 with n > 0, a(0)=0. - Vincenzo Librandi, Nov 26 2010
G.f.: 3*x*(1+4*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
a(0)=0, a(1)=3, a(2)=21, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 08 2012
a(n) = n + A226489(n). - Bruno Berselli, Jun 11 2013
Sum_{n>=1} 1/a(n) = tan(Pi/10)*Pi/9 - sqrt(5)*log(phi)/9 + 5*log(5)/18, where phi is the golden ratio (A001622). - Amiram Eldar, May 20 2023
E.g.f.: 3*exp(x)*x*(2 + 5*x)/2. - Elmo R. Oliveira, Dec 24 2024

A254963 a(n) = n(11n + 3)/2.

Original entry on oeis.org

0, 7, 25, 54, 94, 145, 207, 280, 364, 459, 565, 682, 810, 949, 1099, 1260, 1432, 1615, 1809, 2014, 2230, 2457, 2695, 2944, 3204, 3475, 3757, 4050, 4354, 4669, 4995, 5332, 5680, 6039, 6409, 6790, 7182, 7585, 7999, 8424, 8860, 9307, 9765, 10234, 10714, 11205, 11707

Offset: 0

Bruno Berselli, Feb 11 2015

This sequence provides the first differences of A254407 and the partial sums of A017473.
Also:
a(n) - n         = A022269(n);
a(n) + n         = n*(11*n+5)/2: 0, 8, 27, 57, 98, 150, 213, 287, ...;
a(n) - 2*n       = A022268(n);
a(n) + 2*n       = n*(11*n+7)/2: 0, 9, 29, 60, 102, 155, 219, 294, ...;
a(n) - 3*n       = n*(11*n-3)/2: 0, 4, 19, 45, 82, 130, 189, 259, ...;
a(n) + 3*n       = A211013(n);
a(n) - 4*n       = A226492(n);
a(n) + 4*n       = A152740(n);
a(n) - 5*n       = A180223(n);
a(n) + 5*n       = n*(11*n+13)/2: 0, 12, 35, 69, 114, 170, 237, 315, ...;
a(n) - 6*n       = A051865(n);
a(n) + 6*n       = n*(11*n+15)/2: 0, 13, 37, 72, 118, 175, 243, 322, ...;
a(n) - 7*n       = A152740(n-1) with A152740(-1) = 0;
a(n) + 7*n       = n*(11*n+17)/2: 0, 14, 39, 75, 122, 180, 249, 329, ...;
a(n) - n*(n-1)/2 = A168668(n);
a(n) + n*(n-1)/2 = A049453(n);
a(n) - n*(n+1)/2 = A202803(n);
a(n) + n*(n+1)/2 = A033580(n).

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

Cf. A008729 and A218530 (seventh column); A017473, A254407.
Cf. similar sequences of the type 4*n^2 + k*n*(n+1)/2: A055999 (k=-7, n>6), A028552 (k=-6, n>2), A095794 (k=-5, n>1), A046092 (k=-4, n>0), A000566 (k=-3), A049450 (k=-2), A022264 (k=-1), A016742 (k=0), A022267 (k=1), A202803 (k=2), this sequence (k=3), A033580 (k=4).
Cf. A069125: (2*n+1)^2 + 3*n*(n+1)/2; A147875: n^2 + 3*n*(n+1)/2.
Cf. A022268, A022269, A049453, A051865, A152740, A168668, A180223, A211013, A226492.

Magma
```
[n*(11*n+3)/2: n in [0..50]];
```

Table[n (11 n + 3)/2, {n, 0, 50}]
LinearRecurrence[{3,-3,1},{0,7,25},50] (* Harvey P. Dale, Mar 25 2018 *)

Maxima
```
makelist(n*(11*n+3)/2, n, 0, 50);
```
PARI
```
vector(50, n, n--; n*(11*n+3)/2)
```
Sage
```
[n*(11*n+3)/2 for n in (0..50)]
```

G.f.: x*(7 + 4*x)/(1 - x)^3.
From Elmo R. Oliveira, Dec 15 2024: (Start)
E.g.f.: exp(x)*x*(14 + 11*x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A139272 a(n) = n(8n-5).

Original entry on oeis.org

0, 3, 22, 57, 108, 175, 258, 357, 472, 603, 750, 913, 1092, 1287, 1498, 1725, 1968, 2227, 2502, 2793, 3100, 3423, 3762, 4117, 4488, 4875, 5278, 5697, 6132, 6583, 7050, 7533, 8032, 8547, 9078, 9625, 10188, 10767, 11362, 11973, 12600

Offset: 0

Omar E. Pol, Apr 26 2008

Sequence found by reading the line from 0, in the direction 0, 3, ..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the members of A139276 in the same spiral.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250, A072279, A139273, A139274, A139275, A139276, A139278, A139279, A139280, A139281, A139282.

Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=16: see Comments lines of A226492.

s=0;lst={s};Do[s+=n++ +3;AppendTo[lst, s], {n, 0, 7!, 16}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)
Table[n*(8*n -5), {n,0,50}] (* G. C. Greubel, Jul 18 2017 *)
LinearRecurrence[{3,-3,1},{0,3,22},50] (* Harvey P. Dale, Jan 13 2024 *)

a(n)=n*(8*n-5) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 8*n^2 - 5*n.
Sequences of the form a(n) = 8*n^2 + c*n have generating functions x{c+8+(8-c)x} / (1-x)^3 and recurrence a(n) = 3a(n-1) - 3a(n-2) + a(n-3). The inverse binomial transform is 0, c+8, 16, 0, 0, ... (0 continued). This applies to A139271-A139278, positive or negative c. - R. J. Mathar, May 12 2008
a(n) = 16*n + a(n-1) - 13 with n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
From G. C. Greubel, Jul 18 2017: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(13*x + 3)/(1-x)^3.
E.g.f.: (8*x^2 + 3*x)*exp(x). (End)
Sum_{n>=1} 1/a(n) = ((sqrt(2)-1)*Pi + 8*log(2) - 2*sqrt(2)*log(sqrt(2)+1))/10. - Amiram Eldar, Mar 17 2022

A152759 3 times 9-gonal (or nonagonal) numbers: a(n) = 3n(7*n-5)/2.

Original entry on oeis.org

0, 3, 27, 72, 138, 225, 333, 462, 612, 783, 975, 1188, 1422, 1677, 1953, 2250, 2568, 2907, 3267, 3648, 4050, 4473, 4917, 5382, 5868, 6375, 6903, 7452, 8022, 8613, 9225, 9858, 10512, 11187, 11883, 12600, 13338, 14097, 14877, 15678, 16500, 17343, 18207, 19092, 19998

Offset: 0

Omar E. Pol, Dec 14 2008

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

Cf. A001106, A139268, A226491.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152767, A153783, A153448, A153875.
Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=21: see Comments lines of A226492.

s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,3,6!,21}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2009 *)
CoefficientList[Series[3 x (1 + 6 x) / (1 - x)^3, {x, 0, 60}], x] (* Vincenzo Librandi, Jun 05 2013 *)
LinearRecurrence[{3,-3,1},{0,3,27},40] (* Harvey P. Dale, May 26 2015 *)

a(n)=3*n*(7*n-5)/2 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = (21*n^2 - 15*n)/2 = 3*A001106(n).
a(n) = a(n-1) + 21*n - 18 with n > 0, a(0)=0. - Vincenzo Librandi, Nov 26 2010
G.f.: 3*x*(1+6*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
a(n) = n + A226491(n). - Bruno Berselli, Jun 11 2013
From Elmo R. Oliveira, Dec 15 2024: (Start)
E.g.f.: 3*exp(x)*x*(2 + 7*x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A152767 3 times 10-gonal (or decagonal) numbers: a(n) = 3n(4*n-3).

Original entry on oeis.org

0, 3, 30, 81, 156, 255, 378, 525, 696, 891, 1110, 1353, 1620, 1911, 2226, 2565, 2928, 3315, 3726, 4161, 4620, 5103, 5610, 6141, 6696, 7275, 7878, 8505, 9156, 9831, 10530, 11253, 12000, 12771, 13566, 14385, 15228, 16095, 16986, 17901, 18840, 19803, 20790, 21801

Offset: 0

Omar E. Pol, Dec 15 2008

3*A172078(n) = n*a(n) - Sum_{k=0..n-1} a(k). - Bruno Berselli, Dec 12 2010

			For n=8, a(8) = (1*3 + 5*7 + 9*11 +..+ 29*31) - (2*4 + 6*8 + 10*12 +..+ 26*28) = 696 (see Problem 1052 in References). - _Bruno Berselli_, Dec 12 2010

"Supplemento al Periodico di Matematica", Raffaello Giusti Editore (Livorno), Jan. 1910 p. 47 (Problem 1052).

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

Cf. A001107, A001539, A002939, A139271, A153794, A172078.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152759, A153783, A153448, A153875.
Cf. numbers of the form n*(n*k - k + 6)/2, this sequence is the case k=24: see Comments lines of A226492.

Table[3n(4n-3),{n,0,40}] (* Harvey P. Dale, May 26 2012 *)

a(n)=3*n*(4*n-3) \\ Charles R Greathouse IV, Sep 28 2015

a(n) = 12*n^2 - 9*n = 3*A001107(n).
a(n) = a(n-1) + 24*n - 21, n > 0. - Vincenzo Librandi, Nov 26 2010
a(n) = Sum_{k=0..n-1} A001539(k) - Sum_{k=0..n-1} 4*A002939(k) if n > 0 (see References, Problem 1052). - Bruno Berselli, Dec 08 2010 - Jan 21 2011
G.f.: -3*x*(1+7*x)/(x-1)^3.
a(0)=0, a(1)=3, a(2)=30, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 26 2012
From Elmo R. Oliveira, Dec 15 2024: (Start)
E.g.f.: 3*exp(x)*x*(1 + 4*x).
a(n) = A153794(n) - n. (End)

A153448 3 times 12-gonal (or dodecagonal) numbers: a(n) = 3n(5*n-4).

Original entry on oeis.org

0, 3, 36, 99, 192, 315, 468, 651, 864, 1107, 1380, 1683, 2016, 2379, 2772, 3195, 3648, 4131, 4644, 5187, 5760, 6363, 6996, 7659, 8352, 9075, 9828, 10611, 11424, 12267, 13140, 14043, 14976, 15939, 16932, 17955, 19008, 20091, 21204

Offset: 0

Omar E. Pol, Dec 26 2008

This sequence is related to A172117 by 3*A172117(n) = n*a(n) - Sum_{i=0..n-1} a(i) and this is the case d=10 in the identity n*(3*n*(d*n - d + 2)/2) - Sum_{k=0..n-1} 3*k*(d*k - d + 2)/2 = n*(n+1)*(2*d*n - 2*d + 3)/2. - Bruno Berselli, Aug 26 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A051624, A152965.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152759, A152767, A153783, A153875, A172117.
Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=30: see Comments lines of A226492.

Table[3n(5n-4),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,3,36},40] (* Harvey P. Dale, Jun 18 2014 *)
3*PolygonalNumber[12,Range[0,60]] (* Harvey P. Dale, May 13 2022 *)

a(n)=3*n*(5*n-4) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 15*n^2 - 12*n = A051624(n)*3.
a(n) = 30*n + a(n-1) - 27 with n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: 3*x*(1 + 9*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=3, a(2)=36. - Harvey P. Dale, Jun 18 2014
E.g.f.: 3*x*(1 + 5*x)*exp(x). - G. C. Greubel, Aug 21 2016
a(n) = (4*n-2)^2 - (n-2)^2. In general, if P(k,n) is the k-th n-gonal number, then (2*k+1)*P(8*k+4,n) = ((3*k+1)*n-2*k)^2 - (k*n-2*k)^2. - Charlie Marion, Jul 29 2021

A153783 3 times 11-gonal (or hendecagonal) numbers: a(n) = 3n(9*n-7)/2.

Original entry on oeis.org

0, 3, 33, 90, 174, 285, 423, 588, 780, 999, 1245, 1518, 1818, 2145, 2499, 2880, 3288, 3723, 4185, 4674, 5190, 5733, 6303, 6900, 7524, 8175, 8853, 9558, 10290, 11049, 11835, 12648, 13488, 14355, 15249, 16170, 17118, 18093, 19095

Offset: 0

Omar E. Pol, Jan 02 2009

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

Cf. A051682, A152995.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152759, A152767, A153448, A153875.
Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=27: see Comments lines of A226492.

s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,3,6!,27}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2009 *)
Table[3*n*(9*n-7)/2, {n,0,25}] (* or *) LinearRecurrence[{3,-3,1},{0,3,33},25] (* G. C. Greubel, Aug 28 2016 *)

a(n)=3*n*(9*n-7)/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (27*n^2 - 21*n)/2 = A051682(n)*3.
a(n) = 27*n + a(n-1) - 24, with n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: 3*x*(1 + 8*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
From G. C. Greubel, Aug 28 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: (3/2)*x*(2 + 9*x)*exp(x). (End)

A153875 3 times 13-gonal (or tridecagonal) numbers: a(n) = 3n(11*n - 9)/2.

Original entry on oeis.org

0, 3, 39, 108, 210, 345, 513, 714, 948, 1215, 1515, 1848, 2214, 2613, 3045, 3510, 4008, 4539, 5103, 5700, 6330, 6993, 7689, 8418, 9180, 9975, 10803, 11664, 12558, 13485, 14445, 15438, 16464, 17523, 18615, 19740, 20898, 22089

Offset: 0

Omar E. Pol, Jan 03 2009

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

Cf. A051865, A152997.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152759, A152767, A153783, A153448.
Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=33: see Comments lines of A226492.

Table[(33*n^2 - 27*n)/2, {n,0,25}] (* G. C. Greubel, Aug 31 2016 *)

a(n)=3*n*(11*n-9)/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (33*n^2 - 27*n)/2 = A051865(n)*3.
a(n) = a(n-1) + 33*n - 30, with n>0, a(0)=0. - Vincenzo Librandi, Dec 14 2010
G.f.: 3*x*(1 + 10*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
From G. C. Greubel, Aug 31 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: (3/2)*x*(2 + 11*x)*exp(x). (End)

cp's OEIS Frontend

A094159 3 times hexagonal numbers: a(n) = 3*n*(2*n-1).

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A152751 3 times octagonal numbers: a(n) = 3*n*(3*n-2).

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A152773 3 times heptagonal numbers: a(n) = 3*n*(5*n-3)/2.

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A254963 a(n) = n*(11*n + 3)/2.

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A139272 a(n) = n*(8*n-5).

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A152759 3 times 9-gonal (or nonagonal) numbers: a(n) = 3*n*(7*n-5)/2.

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A152767 3 times 10-gonal (or decagonal) numbers: a(n) = 3*n*(4*n-3).

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A094159 3 times hexagonal numbers: a(n) = 3n(2*n-1).

A152751 3 times octagonal numbers: a(n) = 3n(3*n-2).

A152773 3 times heptagonal numbers: a(n) = 3n(5*n-3)/2.

A254963 a(n) = n(11n + 3)/2.

A139272 a(n) = n(8n-5).

A152759 3 times 9-gonal (or nonagonal) numbers: a(n) = 3n(7*n-5)/2.

A152767 3 times 10-gonal (or decagonal) numbers: a(n) = 3n(4*n-3).

A153448 3 times 12-gonal (or dodecagonal) numbers: a(n) = 3n(5*n-4).

A153783 3 times 11-gonal (or hendecagonal) numbers: a(n) = 3n(9*n-7)/2.

A153875 3 times 13-gonal (or tridecagonal) numbers: a(n) = 3n(11*n - 9)/2.