A049450 -id:A049450 - cp's OEIS frontend

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

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)

A152743 6 times pentagonal numbers: a(n) = 3n(3*n-1).

Original entry on oeis.org

0, 6, 30, 72, 132, 210, 306, 420, 552, 702, 870, 1056, 1260, 1482, 1722, 1980, 2256, 2550, 2862, 3192, 3540, 3906, 4290, 4692, 5112, 5550, 6006, 6480, 6972, 7482, 8010, 8556, 9120, 9702, 10302, 10920, 11556, 12210, 12882, 13572, 14280, 15006, 15750, 16512, 17292

Offset: 0

Omar E. Pol, Dec 12 2008

a(n) is also the Wiener index of the windmill graph D(4,n). The windmill graph D(m,n) is the graph obtained by taking n copies of the complete graph K_m with a vertex in common (i.e. a bouquet of n pieces of K_m graphs). The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph. The Wiener index of D(m,n) is (1/2)n(m-1)[(m-1)(2n-1)+1]. For the Wiener indices of D(3,n), D(5,n), and D(6,n) see A033991, A028994, and A180577, respectively. - Emeric Deutsch, Sep 21 2010

a(n+1) gives the number of edges in a hexagon-like honeycomb built from A003215(n) congruent regular hexagons (see link). Example: a hexagon-like honeycomb consisting of 7 congruent regular hexagons has 1 core hexagon inside a perimeter of six hexagons. The perimeter consists of 18 external edges. There are 6 edges shared by the perimeter hexagons. The core hexagon has 6 edges. a(2) is the total number of edges, i.e. 18 + 6 + 6 = 30. - Ivan N. Ianakiev, Mar 10 2015

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Ivan N. Ianakiev, Hexagon-like honeycomb built from regular congruent hexagons.
Eric Weisstein's World of Mathematics, Windmill Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000326, A152734, A152744, A049450, A062741, A033991, A028994, A180577.

[ 3*n*(3*n-1) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

A152743:=n->3*n*(3*n-1); seq(A152743(n), n=0..50); # Wesley Ivan Hurt, Jun 09 2014

Table[3n(3n-1),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,6,30},40] (* Harvey P. Dale, Jun 30 2011 *)
CoefficientList[Series[-6x (2x+1)/(x-1)^3,{x,0,40}],x] (* Robert G. Wilson v, Mar 10 2015 *)

a(n)=3*n*(3*n-1) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 9n^2 - 3n = A000326(n)*6.
a(n) = A049450(n)*3 = A062741(n)*2. - Omar E. Pol, Dec 15 2008
a(n) = a(n-1) + 18*n - 12 (with a(0)=0). - Vincenzo Librandi, Nov 26 2010
G.f.: -((6*x*(2*x+1))/(x-1)^3). - Harvey P. Dale, Jun 30 2011
E.g.f.: 3*x*(2+3*x)*exp(x). - G. C. Greubel, Sep 01 2018
From Amiram Eldar, Feb 27 2022: (Start)
Sum_{n>=1} 1/a(n) = (9*log(3) - sqrt(3)*Pi)/18.
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi*sqrt(3) - 6*log(2))/9. (End)

Converted reference to link by Omar E. Pol, Oct 07 2010

A035006 Number of possible rook moves on an n X n chessboard.

Original entry on oeis.org

0, 8, 36, 96, 200, 360, 588, 896, 1296, 1800, 2420, 3168, 4056, 5096, 6300, 7680, 9248, 11016, 12996, 15200, 17640, 20328, 23276, 26496, 30000, 33800, 37908, 42336, 47096, 52200, 57660, 63488, 69696, 76296, 83300, 90720, 98568, 106856

Offset: 1

Ulrich Schimke (ulrschimke(AT)aol.com)

Obviously A035005(n) = A002492(n-1) + a(n) since Queen = Bishop + Rook. - Johannes W. Meijer, Feb 04 2010

X values of solutions of the equation: (X-Y)^3-2*X*Y=0. Y values are b(n)=2*n*(n-1)^2 (see A181617). - Mohamed Bouhamida, Jul 06 2023

			On a 3 X 3-board, rook has 9*4 moves, so a(3)=36.

E. Bonsdorff, K. Fabel and O. Riihimaa, Schach und Zahl (Chess and numbers), Walter Rau Verlag, Dusseldorf, 1966.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Alexander M. Haupt, Bijective enumeration of rook walks, arXiv:2007.01018 [math.CO], 2020.
M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
M. Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Richard P. Stanley, Bijective Proof Problems, Problem 540 p. 63, (2015).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A033586 (King), A035005 (Queen), A035008 (Knight), A002492 (Bishop) and A049450 (Pawn).

Cf. A006002, A006564, A006566.

[(n-1)*2*n^2: n in [1..40]]; // Vincenzo Librandi, Jun 16 2011

Table[(n-1) 2 n^2,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,8,36,96},40] (* Harvey P. Dale, May 12 2012 *)

a(n) = (n-1)*2*n^2.
a(n) = Sum_{j=1..n} ((n+j-1)^2 - (n-j+1)^2). - Zerinvary Lajos, Sep 13 2006
1/a(n+1) = Integral_{x=1/(n+1)..1/n} x*h(x) = Integral_{x=1/(n+1)..1/n} x*(1/x - floor(1/x)) = 1/((2*(n^2+2*n+1))*n) and Sum_{n>=1} 1/((2*(n^2+2*n+1))*n) = 1-Zeta(2)/2 where h(x) is the Gauss (continued fraction) map h(x)={x^-1} and {x} is the fractional part of x. - Stephen Crowley, Jul 24 2009
a(n) = 4 * A006002(n-1). - Johannes W. Meijer, Feb 04 2010
G.f.: 4*x^2*(2+x)/(1-x)^4. - Colin Barker, Mar 11 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(1)=0, a(2)=8, a(3)=36, a(4)=96. - Harvey P. Dale, May 12 2012
a(n) = A006566(n) - A006564(n). - Peter M. Chema, Feb 10 2016
E.g.f.: 2*exp(x)*x^2*(2 + x). - Stefano Spezia, May 10 2022
From Amiram Eldar, May 14 2022: (Start)
Sum_{n>=2} 1/a(n) = 1 - Pi^2/12.
Sum_{n>=2} (-1)^n/a(n) = Pi^2/24 + log(2) - 1. (End)

A144390 a(n) = 3*n^2 - n - 1.

Original entry on oeis.org

1, 9, 23, 43, 69, 101, 139, 183, 233, 289, 351, 419, 493, 573, 659, 751, 849, 953, 1063, 1179, 1301, 1429, 1563, 1703, 1849, 2001, 2159, 2323, 2493, 2669, 2851, 3039, 3233, 3433, 3639, 3851, 4069, 4293, 4523, 4759, 5001, 5249, 5503, 5763, 6029, 6301, 6579

Offset: 1

Paul Curtz, Oct 02 2008

Sequence's original Name was "First bisection of A135370."

The partial sums of this sequence give A081437. - Leo Tavares, Dec 26 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
John Elias, Illustration: Belted Hexagrams
Leo Tavares, Illustration: Bounded Hexagons
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016933, A049450, A144391.
Cf. A003215, A005408, A016754, A002378.
Cf. A081437 (partial sums).

[3*n^2-n-1: n in [1..60]]; // Vincenzo Librandi, Jun 14 2011

A144390:=n->3*n^2-n-1; seq(A144390(n), n=1..50); # Wesley Ivan Hurt, Mar 26 2014

Table[3*n^2 -n -1 , {n,0,50}] (* G. C. Greubel, Jul 19 2017 *)

a(n)=3*n^2-n-1 \\ Charles R Greathouse IV, Oct 07 2015

a(n+1) = a(n) + 6*n + 2; see A016933.
G.f.: x*(1+6*x-x^2)/(1-x)^3. a(n) = A049450(n)-1. - R. J. Mathar, Oct 24 2008
a(-n) = A144391(n). - Michael Somos, Mar 27 2014
E.g.f.: (3*x^2 + 2*x -1)*exp(x) + 1. - G. C. Greubel, Jul 19 2017
From Leo Tavares, Dec 26 2021: (Start)
a(n) = A003215(n) - 2*A005408(n). See Bounded Hexagons illustration.
a(n) = A016754(n-1) - A002378(n-2). (End)
a(n) = A003154(n) - A049451(n-1). - John Elias, Dec 22 2022

Edited by R. J. Mathar, Oct 24 2008

More terms from Vladimir Joseph Stephan Orlovsky, Oct 25 2008

A035005 Number of possible queen moves on an n X n chessboard.

Original entry on oeis.org

0, 12, 56, 152, 320, 580, 952, 1456, 2112, 2940, 3960, 5192, 6656, 8372, 10360, 12640, 15232, 18156, 21432, 25080, 29120, 33572, 38456, 43792, 49600, 55900, 62712, 70056, 77952, 86420, 95480, 105152, 115456, 126412, 138040, 150360

Offset: 1

Ulrich Schimke (ulrschimke(AT)aol.com)

The number of (2 to n) digit sequences that can be found reading in any orientation, including diagonals, in an (n X n) grid. - Paul Cleary, Aug 12 2005

			3 X 3 board: queen has 8*6 moves and 1*8 moves, so a(3)=56.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A033586 (King), A035006 (Rook), A035008 (Knight), A002492 (Bishop) and A049450 (Pawn).

Cf. A162147.

[(n-1)*2*n^2 + (4*n^3-6*n^2+2*n)/3: n in [1..40]]; // Vincenzo Librandi, Jun 16 2011

Table[(n-1)2n^2+(4n^3-6n^2+2n)/3,{n,40}] (* or *) LinearRecurrence[ {4,-6,4,-1},{0,12,56,152},40] (* Harvey P. Dale, Aug 24 2011 *)

a(n) = (n-1)*2*n^2 + (4*n^3-6*n^2+2*n)/3.
From Johannes W. Meijer, Feb 04 2010: (Start)
a(n) = A002492(n-1) + A035006(n) since Queen = Bishop + Rook.
a(n) = 4 * A162147(n-1). (End)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=0, a(1)=12, a(2)=56, a(3)=152. - Harvey P. Dale, Aug 24 2011
From Colin Barker, Mar 11 2012: (Start)
a(n) = 2*n*(1-6*n+5*n^2)/3.
G.f.: 4*x^2*(3+2*x)/(1-x)^4. (End)
E.g.f.: 2*exp(x)*x^2*(9 + 5*x)/3. - Stefano Spezia, Jul 31 2022

More terms from Erich Friedman

A307011 First coordinate in a redundant hexagonal coordinate system of the points of a counterclockwise spiral on an hexagonal grid. Second and third coordinates are given in A307012 and A345978.

Original entry on oeis.org

0, 1, 0, -1, -1, 0, 1, 2, 2, 1, 0, -1, -2, -2, -2, -1, 0, 1, 2, 3, 3, 3, 2, 1, 0, -1, -2, -3, -3, -3, -3, -2, -1, 0, 1, 2, 3, 4, 4, 4, 4, 3, 2, 1, 0, -1, -2, -3, -4, -4, -4, -4, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5

Offset: 0

Hugo Pfoertner, Mar 19 2019

From Peter Munn, Jul 22 2021: (Start)
The points of the spiral are equally the points of a hexagonal lattice, the points of an isometric (triangular) grid and the center points of the cells of a honeycomb (regular hexagonal tiling or grid). The coordinate system can be described using 3 axes that pass through spiral point 0 and one of points 1, 2 or 3. Along each axis, one of the coordinates is 0.
a(n) is the signed distance from spiral point n to the axis that passes through point 2. The distance is measured along either of the lines through point n that are parallel to one of the other 2 axes and the sign is such that point 1 has positive distance.
This coordinate can be paired with either of the other coordinates to form oblique coordinates as described in A307012. Alternatively, all 3 coordinates can be used together, symmetrically, as described in A345978.
There is a negated variant of the 3rd coordinate, which is the conventional sense of this coordinate for specifying (with the 2nd coordinate) the Eisenstein integers that can be the points of the spiral when it is embedded in the complex plane. See A307013.
(End)

Hugo Pfoertner, Table of n, a(n) for n = 0..10034
Margherita Barile, Oblique Coordinates, entry in Eric Weisstein's World of Mathematics.
HandWiki, Hexagonal Lattice.
Peter Munn, Illustration of signed distance of spiral points.
Hugo Pfoertner, Illustration of A307012 vs A307011, spiral.
Hugo Pfoertner, Illustration of A345978 vs A307011, spiral.
Wikipedia, Signed distance function.

Cf. A307012, A307013, A307014, A307016, A307017, A328818, A345978.
Numbers on the spokes of the spiral: A000567, A028896, A033428, A045944, A049450, A049451.
Positions on the spiral that correspond to Eisenstein primes: A345435.

r=-1;d=-1;print1(m=0,", ");for(k=0,8,for(j=1,r,print1(s,", "));if(k%2,,m++;r++);for(j=-m,m+1,if(d*j>=-m,print1(s=d*j,", ")));d=-d)

Name revised by Peter Munn, Jul 08 2021

A258222 A(n,k) is the sum over all Dyck paths of semilength n of products over all peaks p of (k*x_p+y_p)/y_p, where x_p and y_p are the coordinates of peak p; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 10, 5, 1, 4, 24, 74, 14, 1, 5, 44, 297, 706, 42, 1, 6, 70, 764, 4896, 8162, 132, 1, 7, 102, 1565, 17924, 100278, 110410, 429, 1, 8, 140, 2790, 47650, 527844, 2450304, 1708394, 1430, 1, 9, 184, 4529, 104454, 1831250, 18685164, 69533397, 29752066, 4862

Offset: 0

Alois P. Heinz, May 23 2015

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

			Square array A(n,k) begins:
:  1,    1,      1,      1,       1,       1, ...
:  1,    2,      3,      4,       5,       6, ...
:  2,   10,     24,     44,      70,     102, ...
:  5,   74,    297,    764,    1565,    2790, ...
: 14,  706,   4896,  17924,   47650,  104454, ...
: 42, 8162, 100278, 527844, 1831250, 4953222, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Wikipedia, Lattice path

Columns k=0-1 give: A000108, A000698(n+1).
Rows n=0-2 give: A000012, A000027(k+1), A049450(k+1).
Main diagonal gives A292694.
Cf. A258219, A258223.

b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (k*x+y)/y, 1)
                 + b(x-1, y+1, true, k)  ))
    end:
A:= (n, k)-> b(2*n, 0, false, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[x_, y_, t_, k_] := b[x, y, t, k] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False, k]*If[t, (k*x + y)/y, 1] + b[x - 1, y + 1, True, k]]];
A [n_, k_] := b[2*n, 0, False, k];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Apr 23 2016, translated from Maple *)

A(n,k) = Sum_{i=0..min(n,k)} C(k,i) * i! * A258223(n,i).

A027599 a(n) = 3n^2 - 7n + 6.

Original entry on oeis.org

6, 2, 4, 12, 26, 46, 72, 104, 142, 186, 236, 292, 354, 422, 496, 576, 662, 754, 852, 956, 1066, 1182, 1304, 1432, 1566, 1706, 1852, 2004, 2162, 2326, 2496, 2672, 2854, 3042, 3236, 3436, 3642, 3854, 4072, 4296, 4526, 4762, 5004, 5252, 5506

Offset: 0

Patrick De Geest

Encyclopaedia Britannica, 1965 ed., Vol. 16 pp. 755-756.

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. A049450.

[((6*n-7)^2+23)/12: n in [0..60]]; // G. C. Greubel, Aug 24 2022

LinearRecurrence[{3,-3,1}, {6,2,4}, 61] (* G. C. Greubel, Aug 24 2022 *)

a(n)=3*n^2-7*n+6 \\ Charles R Greathouse IV, Jun 17 2017

[3*n^2-7*n+6 for n in (0..60)] # G. C. Greubel, Aug 24 2022

a(n) = a(n-1) + 6*n - 10 (with a(0)=6). - Vincenzo Librandi, Nov 19 2010
From Colin Barker, May 22 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: 2*(3 - 8*x + 8*x^2)/(1 - x)^3. (End)
E.g.f.: (6 - 4*x + 3*x^2)*exp(x). - G. C. Greubel, Aug 24 2022

A069743 Let M_n be the n X n matrix M_(i,j)=1/(3^i+3^j), then a(n) is the numerator of det(M_n).

Original entry on oeis.org

1, 1, 1, 169, 57122, 1130708969104, 60520841316555286464512, 967474236461016996630647788281821986816, 3959258211397422699939531791736812415390620457773645692928

Offset: 1

Benoit Cloitre, Apr 21 2002

Curiously, sequence seems related to pentagonal (or 5-gonal) or heptagonal (or 7-gonal) numbers. Some primes follow rules in a(n) factorization. If b(n)= exponent of 13 in a(n) factorization: b(n)=0, 0, 0, 2, 4, 6, 10, 14, 18, 24, 30, 36, 44, 52, 60, 70, 80, 90...so b(3n+1)=A049450(n); b(3n+2)=A049450(n)+2*n; b(3n+3)=A049450(n)+4n. If c(n)= exponent of 11 in a(n) factorization: c(n)=4*(0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 18, 21, 24, 27, ..) so c(5n+1)=4*A000566(n); c(5n+2)=4*(A000566(n)+2n); c(5n+3)=4*(A000566(n)+3n); c(5n+4)=4*(A000566(n)+4n); c(5n+5)=4*(A000566(n)+5n)

Robert Israel, Table of n, a(n) for n = 1..21

Cf. A000566, A049450, A379913.

f:= proc(n) local M;
  M:= Matrix(n,n,(i,j) -> 1/(3^i+3^j));
  numer(LinearAlgebra:-Determinant(M))
end proc:
map(f, [$1..10]); # Robert Israel, Jan 06 2025

for(n=1,15,print1((numerator(matdet(matrix(n,n,i,j,1/(3^j+3^i))))),","))

A153792 12 times pentagonal numbers: a(n) = 6n(3*n-1).

Original entry on oeis.org

0, 12, 60, 144, 264, 420, 612, 840, 1104, 1404, 1740, 2112, 2520, 2964, 3444, 3960, 4512, 5100, 5724, 6384, 7080, 7812, 8580, 9384, 10224, 11100, 12012, 12960, 13944, 14964, 16020, 17112, 18240, 19404, 20604, 21840, 23112, 24420

Offset: 0

Omar E. Pol, Jan 01 2009

For n>=1, a(n) is the first Zagreb index of the triangular grid graph T[n] (see the West reference, p. 390). The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph. - Emeric Deutsch, Nov 10 2016
The M-polynomial of the triangular grid graph T[n] is M(T[n], x, y) = 6*x^2*y^4 + 3*(n-1)*x^4*y^4 +6*(n-2)*x^4*y^6+3*(n-2)*(n-3)*x^6*y^6/2. - Emeric Deutsch, May 09 2018
This is the number of overlapping six sphinx tiled shapes in the sphinx tessellated hexagon described in A291582. - Craig Knecht, Sep 13 2017
a(n) is the number of words of length 3n over the alphabet {a,b,c}, where the number of b's plus the number of c's is 2. - Juan Camacho, Mar 03 2021
Sequence found by reading the line from 0, in the direction 0, 12, ..., in the square spiral whose vertices are the generalized 11-gonal numbers A195160. - Omar E. Pol, Mar 12 2021

D. B. West, Introduction to Graph Theory, 2nd edition, Prentice-Hall, 2001.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Emeric Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
Craig Knecht, Example of 12 overlapping shapes in the order 1 hexagon.
Eric Weisstein's World of Mathematics, Triangular Grid Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000326, A049450, A062741, A033579, A152743, A153449, A153793.

Cf. A008588, A195160, A195321.

List([0..50],n->6*n*(3*n-1)); # Muniru A Asiru, May 10 2018

seq(6*n*(3*n-1),n=0..50); # Robert Israel, Nov 10 2016

Table[6n(3n-1),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,12,60},40] (* Harvey P. Dale, Mar 11 2012 *)

a(n)=6*n*(3*n-1) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 18*n^2 - 6*n = 12*A000326(n) = 6*A049450(n) = 4*A062741(n) = 3*A033579(n) = 2*A152743(n).
a(n) = 36*n + a(n-1) - 24 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
G.f.: 12*x*(1 + 2*x)/(1-x)^3. - Colin Barker, Feb 14 2012
a(0)=0, a(1)=12, a(2)=60; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Mar 11 2012
E.g.f.: 6*x*(2 + 3*x)*exp(x). - G. C. Greubel, Aug 29 2016
a(n) = A291582(n) - A195321(n) for n > 0. - Craig Knecht, Sep 13 2017
a(n) = A195321(n) - A008588(n). - Omar E. Pol, Mar 12 2021
From Amiram Eldar, May 05 2025: (Start)
Sum_{n>=1} 1/a(n) = log(3)/4 - Pi/(12*sqrt(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(6*sqrt(3)) - log(2)/3. (End)

cp's OEIS Frontend

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

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A152743 6 times pentagonal numbers: a(n) = 3*n*(3*n-1).

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A035006 Number of possible rook moves on an n X n chessboard.

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A144390 a(n) = 3*n^2 - n - 1.

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A035005 Number of possible queen moves on an n X n chessboard.

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A307011 First coordinate in a redundant hexagonal coordinate system of the points of a counterclockwise spiral on an hexagonal grid. Second and third coordinates are given in A307012 and A345978.

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A258222 A(n,k) is the sum over all Dyck paths of semilength n of products over all peaks p of (k*x_p+y_p)/y_p, where x_p and y_p are the coordinates of peak p; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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

A152743 6 times pentagonal numbers: a(n) = 3n(3*n-1).

A027599 a(n) = 3n^2 - 7n + 6.

A153792 12 times pentagonal numbers: a(n) = 6n(3*n-1).