A082040 -id:A082040 - cp's OEIS frontend

A303739 Numbers k such that 9k^2 + 3k + 1 (A082040) is prime.

1, 2, 4, 5, 7, 8, 9, 11, 18, 19, 22, 23, 25, 26, 30, 33, 35, 37, 39, 46, 47, 49, 50, 51, 54, 56, 63, 64, 77, 82, 93, 96, 103, 112, 114, 116, 117, 119, 123, 126, 127, 134, 135, 138, 142, 145, 149, 151, 152, 163, 165, 175, 177, 179, 180, 189, 193, 194, 201, 203

Offset: 1

Muniru A Asiru, Jun 01 2018

nonn

These are the indices of the primes in A082040.

Cf. A082040, A303740 (corresponding primes).

Filtered([0..300],n->IsPrime(9*n^2+3*n+1));

select(n->isprime(9*n^2+3*n+1),[$0..300]);

A019557 Coordination sequence for G_2 lattice.

Original entry on oeis.org

1, 12, 30, 48, 66, 84, 102, 120, 138, 156, 174, 192, 210, 228, 246, 264, 282, 300, 318, 336, 354, 372, 390, 408, 426, 444, 462, 480, 498, 516, 534, 552, 570, 588, 606, 624, 642, 660, 678, 696, 714, 732, 750, 768, 786, 804, 822, 840, 858, 876, 894, 912, 930, 948, 966, 984, 1002, 1020, 1038, 1056

Offset: 0

Michael Baake (mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de)

Also, coordination sequence of Dual(3.12.12) tiling with respect to a 12-valent node. - N. J. A. Sloane, Jan 22 2018

For n > 1, also the number of minimum vertex colorings of the n-Andrásfai graph. - Eric W. Weisstein, Mar 03 2024

			From _Peter M. Chema_, Mar 20 2016: (Start)
Illustration of initial terms:
                                                       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
                                                       o
     1           12                30                 48
Compare to A003154, A045946, and A270700. (End)

Michael Baake and Uwe Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, Zeit. f. Kristallographie, 212 (1997), pp. 253-256
Roland Bacher, Pierre de la Harpe, and Boris Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Séries 1) (1997), pp. 1137-1142.
Roland Bacher, Pierre de la Harpe, and Boris Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, Annales de l'institut Fourier, 49 no. 3 (1999), pp. 727-762.
Tom Karzes, Tiling Coordination Sequences.
N. J. A. Sloane, Illustration of layers 0,1,2 in the graph of the Dual(3.12.12) tiling. Centered at a 12-valent node. Note that some of the blue edges are not part of the underlying graph.
N. J. A. Sloane, Overview of coordination sequences of Laves tilings. [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
Eric Weisstein's World of Mathematics, Andrásfai Graph.
Eric Weisstein's World of Mathematics, Minimum Vertex Coloring.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000290, A003154, A008706, A016789, A045946, A270700.
For partial sums see A082040.
List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.

CoefficientList[Series[(1 + 10 x + 7 x^2)/(1 - x)^2, {x, 0, 59}], x] (* Michael De Vlieger, Mar 21 2016 *)

my(x='x+O('x^100)); Vec((1+10*x+7*x^2)/(1-x)^2) \\ Altug Alkan, Mar 20 2016

a(n) = 18*n - 6, n >= 1.
G.f.: (1 + 10*x + 7*x^2)/(1-x)^2.
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: 6*exp(x)*(3*x - 1) + 7.
a(n) = 6*A016789(n-1) for n >= 1.
a(n) = 2*a(n-1) - a(n-2) 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

A082039 Symmetric square array defined by T(n,k) = k^2n^2 + kn + 1, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 13, 21, 13, 1, 1, 21, 43, 43, 21, 1, 1, 31, 73, 91, 73, 31, 1, 1, 43, 111, 157, 157, 111, 43, 1, 1, 57, 157, 241, 273, 241, 157, 57, 1, 1, 73, 211, 343, 421, 421, 343, 211, 73, 1, 1, 91, 273, 463, 601, 651, 601, 463, 273, 91, 1, 1, 111, 343, 601, 813, 931, 931, 813, 601, 343, 111, 1

Offset: 0

Paul Barry, Apr 02 2003

			Square array T(n,k) begins:
  1  1  1   1   1   1 ...
  1  3  7  13  21  31 ...
  1  7 21  43  73 111 ...
  1 13 43  91 157 241 ...
  1 21 73 157 273 421 ...
  ...

Rows include A054569, A002061, A082040, A082041.
Main diagonal is A059826.
Cf. A082038.

A304836 a(n) = 27n^2 - 51n + 24, n>=1.

Original entry on oeis.org

0, 30, 114, 252, 444, 690, 990, 1344, 1752, 2214, 2730, 3300, 3924, 4602, 5334, 6120, 6960, 7854, 8802, 9804, 10860, 11970, 13134, 14352, 15624, 16950, 18330, 19764, 21252, 22794, 24390, 26040, 27744, 29502, 31314, 33180, 35100, 37074, 39102, 41184, 43320, 45510, 47754, 50052, 52404

Offset: 1

Emeric Deutsch, May 21 2018

a(n) is the number of edges in the hex derived network HDN1(n) from the Manuel et al. reference (see HDN1(4) in Fig. 8).

Colin Barker, Table of n, a(n) for n = 1..1000
P. Manuel, R. Bharati, I. Rajasingh, and Chris Monica M, On minimum metric dimension of honeycomb networks, J. Discrete Algorithms, 6, 2008, 20-27.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A082040, A304837, A304838.

List([1..50], n->27*n^2-51*n+24); # Muniru A Asiru, May 21 2018

Maple
```
seq(27*n^2-51*n+24, n = 1 .. 45);
```

concat(0, Vec(6*x^2*(5 + 4*x) / (1 - x)^3 + O(x^40))) \\ Colin Barker, May 23 2018

From Colin Barker, May 23 2018: (Start)
G.f.: 6*x^2*(5 + 4*x) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
(End)

A304837 a(n) = 6(n - 1)(81*n - 104) for n >= 1.

Original entry on oeis.org

0, 348, 1668, 3960, 7224, 11460, 16668, 22848, 30000, 38124, 47220, 57288, 68328, 80340, 93324, 107280, 122208, 138108, 154980, 172824, 191640, 211428, 232188, 253920, 276624, 300300, 324948, 350568, 377160, 404724, 433260, 462768, 493248, 524700, 557124, 590520, 624888, 660228, 696540, 733824, 772080, 811308

Offset: 1

Emeric Deutsch, May 21 2018

a(n) is the first Zagreb index of the hex derived network HDN1(n) from the Manuel et al. reference (see HDN1(4) in Fig. 8).
The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums d(i) + d(j) over all edges ij of the graph.
The M-polynomial of HDN1(n) is M(HDN1(n); x, y) = 12*x^3*y^5 + (18*(n-2))*x^3*y^7 + (6*(3*n^2-9*n+7))*x^3*y^12 + 12*x^5*y^7 + 6*x^5*y^12 + (6*(n-3))*x^7*y^7 + (12*(n-2))*x^7*y^12 + (3*(n-2)*(3*n-5)*x^12*y^12.
54*a(n) + 529 is a square. - Bruno Berselli, May 22 2018

Colin Barker, Table of n, a(n) for n = 1..1000
E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
P. Manuel, R. Bharati, I. Rajasingh, and Chris Monica M, On minimum metric dimension of honeycomb networks, J. Discrete Algorithms, 6, 2008, 20-27.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A082040, A304836, A304838.

List([1..50], n->486*n^2-1110*n+624); # Muniru A Asiru, May 22 2018

Maple
```
seq(624-1110*n+486*n^2, n = 1 .. 45);
```

Table[6 (n - 1) (81 n - 104), {n, 1, 50}] (* Bruno Berselli, May 22 2018 *)

concat(0, Vec(12*x^2*(29 + 52*x)/(1 - x)^3 + O(x^40))) \\ Colin Barker, May 23 2018

G.f.: 12*x^2*(29 + 52*x)/(1 - x)^3. - Bruno Berselli, May 22 2018

A304838 a(n) = 1944n^2 - 5016n + 3138 (n >= 1).

Original entry on oeis.org

66, 882, 5586, 14178, 26658, 43026, 63282, 87426, 115458, 147378, 183186, 222882, 266466, 313938, 365298, 420546, 479682, 542706, 609618, 680418, 755106, 833682, 916146, 1002498, 1092738, 1186866, 1284882, 1386786, 1492578, 1602258, 1715826, 1833282, 1954626, 2079858

Offset: 1

Emeric Deutsch, May 21 2018

a(n) is the second Zagreb index of the hex derived network HDN1(n) from the Manuel et al. reference (see HDN1(4) in Fig. 8).
The second Zagreb index of a simple connected graph is the sum of the degree products d(i)d(j) over all edges ij of the graph.
The M-polynomial of HDN1(n) is M(HDN1(n);x,y) = 12*x^3*y^5 + (18*(n-2))*x^3*y^7 + (6*(3*n^2-9*n+7))*x^3*y^12 + 12*x^5*y^7 + 6*x^5*y^12 + (6*(n-3))*x^7*y^7 + (12*(n-2))*x^7*y^12 + (3*(n-2)*(3*n-5)*x^12*y^12.

Colin Barker, Table of n, a(n) for n = 1..1000
E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
P. Manuel, R. Bharati, I. Rajasingh, and Chris Monica M, On minimum metric dimension of honeycomb networks, J. Discrete Algorithms, 6, 2008, 20-27.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A082040, A304836, A304837.

List([1..50], n->1944*n^2-5016*n+3138); # Muniru A Asiru, May 22 2018

seq(3138 - 5016*n + 1944*n^2, n = 1 .. 45);

Table[1944 n^2 - 5016 n + 3138, {n, 1, 40}] (* Bruno Berselli, May 22 2018 *)
LinearRecurrence[{3,-3,1},{66,882,5586},40] (* Harvey P. Dale, Dec 02 2018 *)

Vec(6*x*(11 + 114*x + 523*x^2)/(1 - x)^3 + O(x^40)) \\ Colin Barker, May 23 2018

G.f.: 6*x*(11 + 114*x + 523*x^2)/(1 - x)^3. - Bruno Berselli, May 22 2018

A245557 Irregular triangle read by rows: T(n,k) (n>=0, 0 <= k <= 2n) = number of triples (u,v,w) with entries in the range 0 to n which have some pair adding up to k and in which at least one of u,v,w is equal to n.

Original entry on oeis.org

1, 3, 6, 4, 3, 6, 15, 12, 7, 3, 6, 9, 24, 21, 18, 10, 3, 6, 9, 12, 33, 30, 27, 24, 13, 3, 6, 9, 12, 15, 42, 39, 36, 33, 30, 16, 3, 6, 9, 12, 15, 18, 51, 48, 45, 42, 39, 36, 19, 3, 6, 9, 12, 15, 18, 21, 60, 57, 54, 51, 48, 45, 42, 22

Offset: 0

N. J. A. Sloane, Aug 04 2014

The sum of (left-justified) rows 0 through n gives row n of A245556. For example, the sum of rows 0 thru 2 is 7, 12, 19, 12, 7, which is the n=2 row of A245556.

			Triangle begins:
[1]
[3, 6, 4]
[3, 6, 15, 12, 7]
[3, 6, 9, 24, 21, 18, 10]
[3, 6, 9, 12, 33, 30, 27, 24, 13]
[3, 6, 9, 12, 15, 42, 39, 36, 33, 30, 16]
[3, 6, 9, 12, 15, 18, 51, 48, 45, 42, 39, 36, 19]
[3, 6, 9, 12, 15, 18, 21, 60, 57, 54, 51, 48, 45, 42, 22]
...
Example. Suppose n = 2. We find:
triple count pair-sums 0  1  2  3  4
                       -------------
002      3     0,2     3     3
012      6     1,2,3      6  6  6
112      3     2,3           3  3
022      3     2,4           3     3
122      3     3,4              3  3
222      1     4                   1
                       -------------
Totals:                3  6 15 12  7, which is row 2 of the triangle.

Partial sums of the rows gives A245556.

Row sums are A082040.

Maple
```
See A245556.
```

T(n,k) = 3k (0 <= k <= n-1), T(n,k) = 12n-3k-3 (n <= k <= 2n-1), T(n,2n) = 3n+1.

A303740 Primes of the form 9k^2 + 3k + 1.

Original entry on oeis.org

13, 43, 157, 241, 463, 601, 757, 1123, 2971, 3307, 4423, 4831, 5701, 6163, 8191, 9901, 11131, 12433, 13807, 19183, 20023, 21757, 22651, 23563, 26407, 28393, 35911, 37057, 53593, 60763, 78121, 83233, 95791, 113233, 117307, 121453, 123553, 127807, 136531, 143263

Offset: 1

Muniru A Asiru, Jun 01 2018

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A303739.

Subsequence of A082040.

Filtered(List([0..300],n->9*n^2+3*n+1),IsPrime);

[a: n in [0..200] | IsPrime(a) where a is 9*n^2 +3*n+1 ];// Vincenzo Librandi, Jun 25 2018

select(isprime,[seq(9*n^2+3*n+1,n=0..500)]);

Select[Table[9 n^2 + 3 n + 1, {n, 0, 150}], PrimeQ] (* Vincenzo Librandi, Jun 25 2018 *)

a(n) = A082040(A303739(n)). - Elmo R. Oliveira, May 04 2025

A082041 a(n) = 16n^2 + 4n + 1.

Original entry on oeis.org

1, 21, 73, 157, 273, 421, 601, 813, 1057, 1333, 1641, 1981, 2353, 2757, 3193, 3661, 4161, 4693, 5257, 5853, 6481, 7141, 7833, 8557, 9313, 10101, 10921, 11773, 12657, 13573, 14521, 15501, 16513, 17557, 18633, 19741, 20881, 22053, 23257, 24493

Offset: 0

Paul Barry, Apr 02 2003

Also sequence found by reading the segment (1,21) together with the line from 21, in the direction 21, 73, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Column k=4 of A082039.

Cf. A054569, A082040, A074377.

Table[16n^2+4n+1,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,21,73},50] (* Harvey P. Dale, Sep 28 2024 *)

a(n)=16*n^2+4*n+1 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (-1-18*x-13*x^2)/(x-1)^3 . - R. J. Mathar, Dec 03 2014
From Elmo R. Oliveira, Oct 28 2024: (Start)
E.g.f.: exp(x)*(1 + 20*x + 16*x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

cp's OEIS Frontend

A303739 Numbers k such that 9*k^2 + 3*k + 1 (A082040) is prime.

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A019557 Coordination sequence for G_2 lattice.

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A069131 Centered 18-gonal numbers.

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A082039 Symmetric square array defined by T(n,k) = k^2*n^2 + k*n + 1, read by antidiagonals.

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A304836 a(n) = 27*n^2 - 51*n + 24, n>=1.

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A304837 a(n) = 6*(n - 1)*(81*n - 104) for n >= 1.

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A304838 a(n) = 1944*n^2 - 5016*n + 3138 (n >= 1).

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A245557 Irregular triangle read by rows: T(n,k) (n>=0, 0 <= k <= 2n) = number of triples (u,v,w) with entries in the range 0 to n which have some pair adding up to k and in which at least one of u,v,w is equal to n.

A303739 Numbers k such that 9k^2 + 3k + 1 (A082040) is prime.

A082039 Symmetric square array defined by T(n,k) = k^2n^2 + kn + 1, read by antidiagonals.

A304836 a(n) = 27n^2 - 51n + 24, n>=1.

A304837 a(n) = 6(n - 1)(81*n - 104) for n >= 1.

A304838 a(n) = 1944n^2 - 5016n + 3138 (n >= 1).

A303740 Primes of the form 9k^2 + 3k + 1.

A082041 a(n) = 16n^2 + 4n + 1.