A019557 -id:A019557 - cp's OEIS frontend

A017233 a(n) = 9*n + 6.

6, 15, 24, 33, 42, 51, 60, 69, 78, 87, 96, 105, 114, 123, 132, 141, 150, 159, 168, 177, 186, 195, 204, 213, 222, 231, 240, 249, 258, 267, 276, 285, 294, 303, 312, 321, 330, 339, 348, 357, 366, 375, 384, 393, 402, 411, 420, 429, 438, 447, 456, 465, 474, 483

Offset: 0

David J. Horn and Laura Krebs Gordon (lkg615(AT)verizon.net), 1985

General form: (q*n-1)*q, cf. A017233 (q=3), A098502 (q=4). - Vladimir Joseph Stephan Orlovsky, Feb 16 2009

Numbers whose digital root is 6; that is, A010888(a(n)) = 6. (Ball essentially says that Iamblichus (circa 350) announced that a number equal to the sum of three integers 3*n, 3*n - 1, and 3*n - 2 has 6 as what is now called the number's digital root.) - Rick L. Shepherd, Apr 01 2014

W. W. R. Ball, A Short Account of the History of Mathematics, Sterling Publishing Company, Inc., 2001 (Facsimile Edition) [orig. pub. 1912], pages 110-111.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008585, A008591, A010888, A016789, A017209, A017221, A017233, A019557, A098502.

[9*n+6: n in [0..60]]; // Vincenzo Librandi, Jul 24 2011

Range[6, 1000, 9] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
LinearRecurrence[{2,-1},{6,15},60] (* Harvey P. Dale, Feb 01 2014 *)

a(n)=9*n+6 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 3*(2+x)/(x-1)^2. - R. J. Mathar, Mar 20 2018
Sum_{n>=0} (-1)^n/a(n) = sqrt(3)*Pi/27 - log(2)/9. - Amiram Eldar, Dec 12 2021
E.g.f.: 3*exp(x)*(2 + 3*x). - Stefano Spezia, Dec 07 2024
From Elmo R. Oliveira, Apr 10 2025: (Start)
a(n) = 3*A016789(n) = A019557(n+1)/2.
a(n) = 2*a(n-1) - a(n-2). (End)

A082040 a(n) = 9n^2 + 3n + 1.

Original entry on oeis.org

1, 13, 43, 91, 157, 241, 343, 463, 601, 757, 931, 1123, 1333, 1561, 1807, 2071, 2353, 2653, 2971, 3307, 3661, 4033, 4423, 4831, 5257, 5701, 6163, 6643, 7141, 7657, 8191, 8743, 9313, 9901, 10507, 11131, 11773, 12433, 13111, 13807, 14521, 15253, 16003, 16771, 17557

Offset: 0

Paul Barry, Apr 02 2003

4th row of A082039, case k = 3 of family T(n,k) = k^2*n^2 + k*n + 1.
a(n)^2 = 81*n^4 + 54*n^3 + 27*n^2 + 6*n + 1 = (24*((3*((3*n^2 + n)/2)^2 + ((3*n^2 + n)/2))/2) + 1). Therefore, (a(n)^2 - 1)/24 is a second pentagonal number (A005449) of index number equal to the n-th second pentagonal number. For example, a(30) = 8191 and (8191^2 - 1)/24 = (67092481 - 1)/24 = 2795520, the 1365th second pentagonal number. 1365 is the 30th second pentagonal number. - Raphie Frank, Sep 19 2012
For n >= 1, a(n) is the number of vertices in the hex derived network HDN1(n+1) from the Manuel et al. reference (see HFN1(4) in Fig. 8). - Emeric Deutsch, May 21 2018
4*a(n) - 3 is a square. - Muniru A Asiru, May 24 2018

Muniru A Asiru, Table of n, a(n) for n = 0..5000
Nicolay Avilov, Graphic illustration of sequence members
P. Manuel, R. Bharati, I. Rajasingh, and Chris Monica M, On minimum metric dimension of honeycomb networks, Journal of Discrete Algorithms, Vol. 6 (2008), pp. 20-27.
Leo Tavares, Snowflake illustration.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002061, A005449, A045945, A054569, A082039.
Partial sums of A019557.
Cf. A000290, A003215.

List([0..50], n->9*n^2+3*n+1); # Muniru A Asiru, May 21 2018

seq(9*n^2+3*n+1,n=0..50); # Muniru A Asiru, May 21 2018

a(n)=9*n^2+3*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 18*n + a(n-1) - 6 with n > 0, a(0)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = A045945(n) + 1: subsequence of A002061. - Muniru A Asiru, May 26 2018
a(n) = A003215(n) + 6*A000290(n). - Leo Tavares, Jul 14 2023
From Elmo R. Oliveira, Oct 23 2024: (Start)
G.f.: (1 + 10*x + 7*x^2)/(1 - x)^3.
E.g.f.: (1 + 12*x + 9*x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A008340 Coordination sequence for E_8 lattice.

Original entry on oeis.org

1, 240, 9120, 121680, 864960, 4113840, 14905440, 44480400, 114879360, 265422960, 561403680, 1105317840, 2050966080, 3620750640, 6126497760, 9994133520, 15792541440, 24266930160, 36377039520, 53340513360, 76681767360

Offset: 0

N. J. A. Sloane and J. H. Conway

M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, Zeit. f. Kristallographie, 212 (1997), 253-256.
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
G. Nebe and N. J. A. Sloane, Home page for this lattice
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A019557, A019558, A008397, A008399.

if n = 0 then 1 else 456/7*n^7-120*n^6+312*n^5-120*n^4-48*n^3+240*n^2-624/7*n;

Join[{1},Table[456/7*n^7-120*n^6+312*n^5-120*n^4-48*n^3+ 240*n^2- 624/7*n,{n,20}]] (* Harvey P. Dale, Jul 14 2014 *)

a(n) = if n = 0 then 1 else (456/7)*n^7-120*n^6+312*n^5-120*n^4-48*n^3+240*n^2-(624/7)*n.
Bacher et al. give a g.f.
G.f.: (x^8 +232*x^7 +24508*x^6 +107224*x^5 +133510*x^4 +55384*x^3 +7228*x^2 +232*x +1)/(x -1)^8 = 1 + 240*x* (1+30*x+231*x^2+556*x^3+447*x^4+102*x^5+x^6) /(1-x)^8. [Colin Barker, Sep 26 2012]

The values given by O'Keeffe are incorrect.

A008399 Coordination sequence for E_6 lattice.

Original entry on oeis.org

1, 72, 1062, 6696, 26316, 77688, 189810, 405720, 785304, 1408104, 2376126, 3816648, 5885028, 8767512, 12684042, 17891064, 24684336, 33401736, 44426070, 58187880, 75168252, 95901624, 120978594, 151048728, 186823368, 229078440, 278657262, 336473352, 403513236

Offset: 0

N. J. A. Sloane and J. H. Conway

M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, 1997; Zeit. f. Kristallographie, 212 (1997), 253-256.
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A008397, A008340, A019557, A019558.

[1] cat [9*n*(13*n^2+7)*(n^2+1)/5: n in [1..40]]; // G. C. Greubel, May 29 2023

1, seq(117/5*n^5+36*n^3+63/5*n, n=1..30);

LinearRecurrence[{6,-15,20,-15,6,-1},{1,72,1062,6696,26316,77688, 189810},30] (* Harvey P. Dale, Oct 24 2022 *)

[9*n*(13*n^2+7)*(n^2+1)//5 +int(n==0) for n in range(41)] # G. C. Greubel, May 29 2023

a(n) = 9*n*(13*n^2+7)*(n^2+1)/5 for n >= 1.
Bacher et al. give a g.f.
G.f.: (1+66*x+645*x^2+1384*x^3+645*x^4+66*x^5+x^6)/(1-x)^6 = 1 + 18*x*(4+35*x+78*x^2+35*x^3+4*x^4)/(1-x)^6. - Colin Barker, Sep 26 2012
E.g.f.: 1 + (1/5)*x*(360 + 2295*x + 3105*x^2 + 1170*x^3 + 117*x^4 )*exp(x). - G. C. Greubel, May 29 2023

A323183 Consider the family of configurations E where E(0) consists of a single equilateral triangle, and for any k >= 0, E(k+1) is obtained by applying the Equithirds substitution to E(k). For k >= 5, the central node of E(k) has 6 equivalent tetravalent neighbors; let t(k) be the coordination sequence for one of those tetravalent nodes. This sequence is the limit of t(k) as k goes to infinity.

Original entry on oeis.org

1, 4, 20, 39, 55, 71, 91, 107, 129, 147, 165, 181, 197, 217, 233, 253, 269, 289, 305, 325, 341, 361, 377, 399, 417, 435, 453, 471, 489, 507, 525, 543, 559, 575, 595, 611, 631, 647, 667, 683, 703, 719, 739, 755, 775, 791, 811, 827, 847, 863, 883, 899, 919, 935

Offset: 0

Rémy Sigrist, Jan 06 2019

nonn

The variant relative to the central node appears to match A019557.

Rémy Sigrist, Illustration of initial terms
Rémy Sigrist, C# program for A323183
Tilings Encyclopedia, Equithirds
Index entries for sequences related to coordination sequences

Cf. A019557, A323187 (partial sums).

cp's OEIS Frontend

A017233 a(n) = 9*n + 6.

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A082040 a(n) = 9*n^2 + 3*n + 1.

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A008340 Coordination sequence for E_8 lattice.

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A008399 Coordination sequence for E_6 lattice.

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A082040 a(n) = 9n^2 + 3n + 1.