A174709 -id:A174709 - cp's OEIS frontend

A000567 Octagonal numbers: n(3n-2). Also called star numbers.

0, 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, 408, 481, 560, 645, 736, 833, 936, 1045, 1160, 1281, 1408, 1541, 1680, 1825, 1976, 2133, 2296, 2465, 2640, 2821, 3008, 3201, 3400, 3605, 3816, 4033, 4256, 4485, 4720, 4961, 5208, 5461

Offset: 0

N. J. A. Sloane

From Floor van Lamoen, Jul 21 2001: (Start)
Write 1,2,3,4,... in a hexagonal spiral around 0; then a(n) is the sequence found by reading the line from 0 in the direction 0,1,....
The spiral begins:
.
                 85--84--83--82--81--80
                 /                     \
               86  56--55--54--53--52  79
               /   /                 \   \
             87  57  33--32--31--30  51  78
             /   /   /             \   \   \
           88  58  34  16--15--14  29  50  77
           /   /   /   /         \   \   \   \
         89  59  35  17   5---4  13  28  49  76
         /   /   /   /   /     \   \   \   \   \
       90  60  36  18   6   0   3  12  27  48  75
       /   /   /   /   /   /   /   /   /   /   /
     91  61  37  19   7   1---2  11  26  47  74
       \   \   \   \   \ .       /   /   /   /
       92  62  38  20   8---9--10  25  46  73
         \   \   \   \ .           /   /   /
         93  63  39  21--22--23--24  45  72
           \   \   \ .               /   /
           94  64  40--41--42--43--44  71
             \   \ .                   /
             95  65--66--67--68--69--70
               \ .
               96
.
(End)
From Lekraj Beedassy, Oct 02 2003: (Start)
Also the number of distinct three-cell blocks that may be removed out of A000217(n+1) square cells arranged in a stepping triangular array of side (n+1). A 5-layer triangular array of square cells, for instance, has vertices outlined thus:
  x x
  x x x
  x x x x
  x x x x x
  x x x x x x
  x x x x x x  (End)
First derivative at n of A045991. - Ross La Haye, Oct 23 2004
Starting from n=1, the sequence corresponds to the Wiener index of K_{n,n} (the complete bipartite graph wherein each independent set has n vertices). - Kailasam Viswanathan Iyer, Mar 11 2009
Number of divisors of 24^(n-1) for n > 0 (cf A009968). - J. Lowell, Aug 30 2008
a(n) = A001399(6n-5), number of partitions of 6*n - 5 into parts < 4. For example a(2)=8 and partitions of 6*2 - 5 = 7 into parts < 4 are: [1,1,1,1,1,1,1], [1,1,1,1,1,2],[1,1,1,1,3], [1,1,1,2,2], [1,1,2,3], [1,2,2,2], [1,3,3], [2,2,3]. - Adi Dani, Jun 07 2011
Also, sequence found by reading the line from 0 in the direction 0, 8, ..., and the parallel line from 1 in the direction 1, 21, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Sep 10 2011
Partial sums give A002414. - Omar E. Pol, Jan 12 2013
Generate a Pythagorean triple using Euclid's formula with (n, n-1) to give A,B,C. a(n) = B + (A + C)/2. - J. M. Bergot, Jul 13 2013
The number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 773", based on the 5-celled von Neumann neighborhood. - Robert Price, May 23 2016
For n >= 1, the continued fraction expansion of sqrt(27*a(n)) is [9n-4; {1, 2n-2, 3, 2n-2, 1, 18n-8}]. For n=1, this collapses to [5; {5, 10}]. - Magus K. Chu, Oct 10 2022
a(n)*a(n+1) + 1 = (3n^2 + n - 1)^2. In general, a(n)*a(n+k) + k^2 = (3n^2 + (3k-2)n - k)^2. - Charlie Marion, May 23 2023

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 38.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 19-20.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 123.

T. D. Noe, Table of n, a(n) for n = 0..1000
Raghavendra N. Bhat, Cristian Cobeli, and Alexandru Zaharescu, A lozenge triangulation of the plane with integers, arXiv:2403.10500 [math.NT], 2024.
Francesco Brenti and Paolo Sentinelli, Wachs permutations, Bruhat order and weak order, arXiv:2212.04932 [math.CO], 2022.
Cesar Ceballos and Viviane Pons, The s-weak order and s-permutahedra II: The combinatorial complex of pure intervals, arXiv:2309.14261 [math.CO], 2023. See p. 42.
C. K. Cook and M. R. Bacon, Some polygonal number summation formulas, Fib. Q., 52 (2014), 336-343.
John Elias, Illustration: Hexagonal spiral grids based on generalized octagonal numbers; Illustration: Generalized Square-Octagonal Grids.
John Elias, Illustration: Nesting Cube-frames; Nesting Cube Animation; Nesting-frames Decomposition; Factorial Compartmentalization.
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
Lancelot Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 342.
Milan Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
R. Kemp, On the number of words in the language {w in Sigma* | w = w^R }^2, Discrete Math., 40 (1982), 225-234. See Table 1.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.
Kaie Kubjas, Luca Sodomaco, and Elias Tsigaridas, Exact solutions in low-rank approximation with zeros, arXiv:2010.15636 [math.AG], 2020.
Viktor Levandovskyy, Christoph Koutschan, and Oleksandr Motsak, On Two-generated Non-commutative Algebras Subject to the Affine Relation, arXiv:1108.1108 [cs.SC], 2011.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Omar E. Pol, Illustration of initial terms of A000217, A000290, A000326, A000384, A000566, A000567.
Leo Tavares, Illustration: Square Rays.
Leo Tavares, Illustration: Twin Rectangular Rays.
Leo Tavares, Illustration: Star Rows.
Leo Tavares, Illustration: Split Stars.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Eric Weisstein's World of Mathematics, Octagonal Number.
Eric Weisstein's World of Mathematics, Wiener Index.
Index to sequences related to polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014641, A014642, A014793, A014794, A001835, A016777, A045944, A093563 ((6, 1) Pascal, column m=2). A016921 (differences).
Cf. A005408 (the odd numbers).
Cf. A000290, A000217, A046092, A000384, A002378, A003154.

List([0..50], n -> n*(3*n-2)); # G. C. Greubel, Nov 15 2018

a000567 n = n * (3 * n - 2)  -- Reinhard Zumkeller, Dec 20 2012

[n*(3*n-2) : n in [0..50]]; // Wesley Ivan Hurt, Oct 10 2021

A000567 := proc(n)
    n*(3*n-2) ;
end proc:
seq(A000567(n), n=1..50) ;

Table[n (3 n - 2), {n, 0, 50}] (* Harvey P. Dale, May 06 2012 *)
Table[PolygonalNumber[RegularPolygon[8], n], {n, 0, 43}] (* Arkadiusz Wesolowski, Aug 27 2016 *)
PolygonalNumber[8, Range[0, 20]] (* Eric W. Weisstein, Sep 07 2017 *)
LinearRecurrence[{3, -3, 1}, {1, 8, 21}, {0, 20}] (* Eric W. Weisstein, Sep 07 2017 *)

a(n)=n*(3*n-2) \\ Charles R Greathouse IV, Jun 10 2011

vector(50, n, n--; n*(3*n-2)) \\ G. C. Greubel, Nov 15 2018

# Intended to compute the initial segment of the sequence, not isolated terms.
def aList():
     x, y = 1, 1
     yield 0
     while True:
         yield x
         x, y = x + y + 6, y + 6
A000567 = aList()
print([next(A000567) for i in range(49)]) # Peter Luschny, Aug 04 2019

[n*(3*n-2) for n in range(50)] # Gennady Eremin, Mar 10 2022

[n*(3*n-2) for n in range(50)] # G. C. Greubel, Nov 15 2018

a(n) = n*(3*n-2).
a(n) = (3n-2)*(3n-1)*(3n)/((3n-1) + (3n-2) + (3n)), i.e., (the product of three consecutive numbers)/(their sum). a(1) = 1*2*3/(1+2+3), a(2) = 4*5*6/(4+5+6), etc. - Amarnath Murthy, Aug 29 2002
E.g.f.: exp(x)*(x+3*x^2). - Paul Barry, Jul 23 2003
G.f.: x*(1+5*x)/(1-x)^3. Simon Plouffe in his 1992 dissertation
a(n) = Sum_{k=1..n} (5*n - 4*k). - Paul Barry, Sep 06 2005
a(n) = n + 6*A000217(n-1). - Floor van Lamoen, Oct 14 2005
a(n) = C(n+1,2) + 5*C(n,2).
Starting (1, 8, 21, 40, 65, ...) = binomial transform of [1, 7, 6, 0, 0, 0, ...]. - Gary W. Adamson, Apr 30 2008
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(0)=0, a(1)=1, a(2)=8. - Jaume Oliver Lafont, Dec 02 2008
a(n) = A000578(n) - A007531(n). - Reinhard Zumkeller, Sep 18 2009
a(n) = a(n-1) + 6*n - 5 (with a(0)=0). - Vincenzo Librandi, Nov 20 2010
a(n) = 2*a(n-1) - a(n-2) + 6. - Ant King, Sep 01 2011
a(n) = A000217(n) + 5*A000217(n-1). - Vincenzo Librandi, Nov 20 2010
a(n) = (A185212(n) - 1) / 4. - Reinhard Zumkeller, Dec 20 2012
a(n) = A174709(6n). - Philippe Deléham, Mar 26 2013
a(n) = (2*n-1)^2 - (n-1)^2. - Ivan N. Ianakiev, Apr 10 2013
a(6*a(n) + 16*n + 1) = a(6*a(n) + 16*n) + a(6*n + 1). - Vladimir Shevelev, Jan 24 2014
a(0) = 0, a(n) = Sum_{k=0..n-1} A005408(A051162(n-1,k)), n >= 1. - L. Edson Jeffery, Jul 28 2014
Sum_{n>=1} 1/a(n) = (sqrt(3)*Pi + 9*log(3))/12 = 1.2774090575596367311949534921... . - Vaclav Kotesovec, Apr 27 2016
From Ilya Gutkovskiy, Jul 29 2016: (Start)
Inverse binomial transform of A084857.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(2*sqrt(3)) = A093766. (End)
a(n) = n * A016777(n-1) = A053755(n) - A000290(n+1). - Bruce J. Nicholson, Aug 10 2017
Product_{n>=2} (1 - 1/a(n)) = 3/4. - Amiram Eldar, Jan 21 2021
P(4k+4,n) = ((k+1)*n - k)^2 - (k*n - k)^2 where P(m,n) is the n-th m-gonal number (a generalization of the Apr 10 2013 formula, a(n) = (2*n-1)^2 - (n-1)^2). - Charlie Marion, Oct 07 2021
From Leo Tavares, Oct 31 2021: (Start)
a(n) = A000290(n) + 4*A000217(n-1). See Square Rays illustration.
a(n) = A000290(n) + A046092(n-1)
a(n) = A000384(n) + 2*A000217(n-1). See Twin Rectangular Rays illustration.
a(n) = A000384(n) + A002378(n-1)
a(n) = A003154(n) - A045944(n-1). See Star Rows illustration. (End)

Incorrect example removed by Joerg Arndt, Mar 11 2010

A033428 a(n) = 3*n^2.

Original entry on oeis.org

0, 3, 12, 27, 48, 75, 108, 147, 192, 243, 300, 363, 432, 507, 588, 675, 768, 867, 972, 1083, 1200, 1323, 1452, 1587, 1728, 1875, 2028, 2187, 2352, 2523, 2700, 2883, 3072, 3267, 3468, 3675, 3888, 4107, 4332, 4563, 4800, 5043, 5292, 5547, 5808, 6075, 6348

Offset: 0

Jeff Burch

The number of edges of a complete tripartite graph of order 3n, K_n,n,n. - Roberto E. Martinez II, Oct 18 2001
From Floor van Lamoen, Jul 21 2001: (Start)
Write 1,2,3,4,... in a hexagonal spiral around 0; then a(n) is the sequence found by reading the line from 0 in the direction 0,3,.... The spiral begins:
.
            33--32--31--30
            /             \
          34  16--15--14  29
          /   /         \   \
        35  17   5---4  13  28
        /   /   /     \   \   \
      36  18   6   0---3--12--27--48-->
      /   /   /   /   /   /   /   /
    37  19   7   1---2  11  26  47
      \   \   \         /   /   /
      38  20   8---9--10  25  46
        \   \             /   /
        39  21--22--23--24  45
          \                 /
          40--41--42--43--44
(End)
Number of edges of the complete bipartite graph of order 4n, K_n,3n. - Roberto E. Martinez II, Jan 07 2002
Also the number of partitions of 6n + 3 into at most 3 parts. - R. K. Guy, Oct 23 2003
Also the number of partitions of 6n into exactly 3 parts. - Colin Barker, Mar 23 2015
Numbers n such that the imaginary quadratic field Q[sqrt(-n)] has six units. - Marc LeBrun, Apr 12 2006
The denominators of Hoehn's sequence (recalled by G. L. Honaker, Jr.) and the numerators of that sequence reversed. The sequence is 1/3, (1+3)/(5+7), (1+3+5)/(7+9+11), (1+3+5+7)/(9+11+13+15), ...; reduced to 1/3, 4/12, 9/27, 16/48, ... . For the reversal, the reduction is 3/1, 12/4, 27/9, 48/16, ... . - Enoch Haga, Oct 05 2007
Right edge of tables in A200737 and A200741: A200737(n, A000292(n)) = A200741(n, A100440(n)) = a(n). - Reinhard Zumkeller, Nov 21 2011
The Wiener index of the crown graph G(n) (n>=3). The crown graph G(n) is the graph with vertex set {x(1), x(2), ..., x(n), y(1), y(2), ..., y(n)} and edge set {(x(i), y(j)): 1<=i, j<=n, i/=j} (= the complete bipartite graph K(n,n) with horizontal edges removed). Example: a(3)=27 because G(3) is the cycle C(6) and 6*1 + 6*2 + 3*3 = 27. The Hosoya-Wiener polynomial of G(n) is n(n-1)(t+t^2)+nt^3. - Emeric Deutsch, Aug 29 2013
From Michel Lagneau, May 04 2015: (Start)
Integer area A of equilateral triangles whose side lengths are in the commutative ring Z[3^(1/4)] = {a + b*3^(1/4) + c*3^(1/2) + d*3^(3/4), a,b,c and d in Z}.
The area of an equilateral triangle of side length k is given by A = k^2*sqrt(3)/4. In the ring Z[3^(1/4)], if k = q*3^(1/4), then A = 3*q^2/4 is an integer if q is even. Example: 27 is in the sequence because the area of the triangle (6*3^(1/4), 6*3^(1/4), 6*3^(1/4)) is 27. (End)
a(n) is 2*sqrt(3) times the area of a 30-60-90 triangle with short side n. Also, 3 times the area of an n X n square. - Wesley Ivan Hurt, Apr 06 2016
Consider the hexagonal tiling of the plane. Extract any four hexagons adjacent by edge. This can be done in three ways. Fold the four hexagons so that all opposite faces occupy parallel planes. For all parallel projections of the resulting object, at least two correspond to area a(n) for side length of n of the original hexagons. - Torlach Rush, Aug 17 2022
The sequence terms are the exponents in the expansion of Product_{n >= 1} (1 - q^(3*n))/(1 + q^(3*n)) = ( Sum_{n in Z} q^(n*(3*n+1)/2) ) / ( Product_{n >= 1} 1 + q^n ) = 1 - 2*q^3 + 2*q^12 - 2*q^27 + 2*q^48 - 2*q^75 + - .... - Peter Bala, Dec 30 2024

			From _Ilya Gutkovskiy_, Apr 13 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
. n=1      n=2            n=3                 n=4
(End)

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
Francesco Brenti and Paolo Sentinelli, Wachs permutations, Bruhat order and weak order, arXiv:2212.04932 [math.CO], 2022.
A. J. C. Cunningham, Factorisation of N and N' = (x^n -+ y^n) / (x -+ y) [when x-y=n], Messenger Math., 54 (1924), 17-21. [Incomplete annotated scanned copy]
Frank Ellermann, Illustration of binomial transforms.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Leo Tavares, Hexagonal illustration
Eric Weisstein's World of Mathematics, Crown Graph.
Eric Weisstein's World of Mathematics, Wiener Index.
Eric Weisstein's World of Mathematics, Unit.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000567, A000217, A000290, A033581, A033583, A092205, A092206.
3 times n-gonal numbers: A045943, A062741, A094159, A152773, A152751, A152759, A152767, A153783, A153448, A153875.
Cf. A219056.
Cf. A000326, A005449, A086463.
Cf. A003215, A016777.

a033428 = (* 3) . (^ 2)
a033428_list = 0 : 3 : 12 : zipWith (+) a033428_list
   (map (* 3) $ tail $ zipWith (-) (tail a033428_list) a033428_list)
-- Reinhard Zumkeller, Jul 11 2013

[3*n^2: n in [0..50]]; // Vincenzo Librandi, May 18 2015

seq(3*n^2, n=0..46); # Nathaniel Johnston, Jun 26 2011

3 Range[0, 50]^2
LinearRecurrence[{3, -3, 1}, {0, 3, 12}, 50] (* Harvey P. Dale, Feb 16 2013 *)

makelist(3*n^2,n,0,30); /* Martin Ettl, Nov 12 2012 */

PARI
```
a(n)=3*n^2
```

def a(n): return 3 * (n**2) # Torlach Rush, Aug 25 2022

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>2.
G.f.: 3*x*(1+x)/(1-x)^3. - R. J. Mathar, Sep 09 2008
Main diagonal of triangle in A132111: a(n) = A132111(n,n). - Reinhard Zumkeller, Aug 10 2007
A214295(a(n)) = -1. - Reinhard Zumkeller, Jul 12 2012
a(n) = A215631(n,n) for n > 0. - Reinhard Zumkeller, Nov 11 2012
a(n) = A174709(6n+2). - Philippe Deléham, Mar 26 2013
a(n) = a(n-1) + 6*n - 3, with a(0)=0. - Jean-Bernard François, Oct 04 2013
E.g.f.: 3*x*(1 + x)*exp(x). - Ilya Gutkovskiy, Apr 13 2016
a(n) = t(3*n) - 3*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): A000217(3*n) - 3*A000217(n). - Bruno Berselli, Aug 31 2017
a(n) = A000326(n) + A005449(n). - Bruce J. Nicholson, Jan 10 2020
From Amiram Eldar, Jul 03 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/18 (A086463).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/36. (End)
From Amiram Eldar, Feb 03 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = sqrt(3)*sinh(Pi/sqrt(3))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(3)*sin(Pi/sqrt(3))/Pi. (End)
a(n) = A003215(n) - A016777(n). - Leo Tavares, Apr 29 2023

Better description from N. J. A. Sloane, May 15 1998

A045944 Rhombic matchstick numbers: a(n) = n(3n+2).

Original entry on oeis.org

0, 5, 16, 33, 56, 85, 120, 161, 208, 261, 320, 385, 456, 533, 616, 705, 800, 901, 1008, 1121, 1240, 1365, 1496, 1633, 1776, 1925, 2080, 2241, 2408, 2581, 2760, 2945, 3136, 3333, 3536, 3745, 3960, 4181, 4408, 4641, 4880, 5125, 5376, 5633, 5896, 6165, 6440

Offset: 0

R. K. Guy

From Floor van Lamoen, Jul 21 2001: (Start)
Write 1,2,3,4,... in a hexagonal spiral around 0, then a(n) is the n-th term of the sequence found by reading the line from 0 in the direction 0,5,.... The spiral begins:
                 .
                 85--84--83--82--81--80
                   .                   \
                   56--55--54--53--52  79
                   / .               \   \
                 57  33--32--31--30  51  78
                 /   / .           \   \   \
               58  34  16--15--14  29  50  77
               /   /   / .       \   \   \   \
             59  35  17   5---4  13  28  49  76
             /   /   /   / .   \   \   \   \   \
           60  36  18   6   0   3  12  27  48  75
           /   /   /   /   /   /   /   /   /   /
         61  37  19   7   1---2  11  26  47  74
           \   \   \   \         /   /   /   /
           62  38  20   8---9--10  25  46  73
             \   \   \             /   /   /
             63  39  21--22--23--24  45  72
               \   \                 /   /
               64  40--41--42--43--44  71
                 \                     /
                 65--66--67--68--69--70
(End)
Connection to triangular numbers: a(n) = 4*T_n + S_n where T_n is the n-th triangular number and S_n is the n-th square. - William A. Tedeschi, Sep 12 2010
Also, second octagonal numbers. - Bruno Berselli, Jan 13 2011
Sequence found by reading the line from 0, in the direction 0, 16, ... and the line from 5, in the direction 5, 33, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012
Let P denote the points from the n X n grid. A(n-1) also coincides with the minimum number of points Q needed to "block" P, that is, every line segment spanned by two points from P must contain one point from Q. - Manfred Scheucher, Aug 30 2018
Also the number of internal edges of an (n+1)*(n+1) "square" of hexagons; i.e., n+1 rows, each of n+1 edge-adjacent hexagons, stacked with minimal overhang. - Jon Hart, Sep 29 2019
For n >= 1, the continued fraction expansion of sqrt(27*a(n)) is [9n+2; {1, 2n-1, 1, 1, 1, 2n-1, 1, 18n+4}]. - Magus K. Chu, Oct 13 2022

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
M. Janjic and B. Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
Leo Tavares, Illustration: Square Stars
Leo Tavares, Illustration: Split Stars
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A001859. See Comments of A135713.
Cf. A000217, A000567, A001082, A002378, A016969, A049450, A174709.
Cf. second n-gonal numbers: A005449, A014105, A147875, A179986, A033954, A062728, A135705.
Cf. numbers of the form n*(d*n+10-d)/2: A008587, A056000, A028347, A140090, A014106, A028895, A186029, A007742, A022267, A033429, A022268, A049452, A186030, A135703, A152734, A139273.
Cf. A000290, A022144, A005563, A001105.
Cf. A056109.
Cf. A003154.

[n*(3*n+2) : n in [0..100]]; // Wesley Ivan Hurt, Sep 24 2017

Table[n*(3n+2), {n,0,60}] (* Harvey P. Dale, May 05 2011 *)
LinearRecurrence[{3,-3,1},{0,5,16},60] (* Harvey P. Dale, Jan 19 2016 *)
CoefficientList[Series[x*(5 + x)/(1 - x)^3,{x, 0, 60}], x] (* Stefano Spezia, Sep 01 2018 *)

a(n)=n*(3*n+2) \\ Charles R Greathouse IV, Nov 20 2012

O.g.f.: x*(5+x)/(1-x)^3. - R. J. Mathar, Jan 07 2008
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(0)=0, a(1)=5, a(2)=16. - Harvey P. Dale, May 06 2011
a(n) = a(n-1) + 6*n - 1 (with a(0)=0). - Vincenzo Librandi, Nov 18 2010
For n > 0, a(n)^3 + (a(n)+1)^3 + ... + (a(n)+n)^3 + 2*A000217(n)^2 = (a(n) + n + 1)^3 + ... + (a(n) + 2n)^3; see also A033954. - Charlie Marion, Dec 08 2007
a(n) = Sum_{i=0..n-1} A016969(i) for n > 0. - Bruno Berselli, Jan 13 2011
a(n) = A174709(6*n+4). - Philippe Deléham, Mar 26 2013
a(n) = A001082(2*n). - Michael Turniansky, Aug 24 2013
Sum_{n>=1} 1/a(n) = (9 + sqrt(3)*Pi - 9*log(3))/12 = 0.3794906245574721941... . - Vaclav Kotesovec, Apr 27 2016
a(n) = A002378(n) + A014105(n). - J. M. Bergot, Apr 24 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/sqrt(12) - 3/4. - Amiram Eldar, Jul 03 2020
E.g.f.: exp(x)*x*(5 + 3*x). - Stefano Spezia, Jun 08 2021
From Leo Tavares, Oct 14 2021: (Start)
a(n) = A000290(n) + 4*A000217(n). See Square Stars illustration.
a(n) = A000567(n+2) - A022144(n+1)
a(n) = A005563(n) + A001105(n).
a(n) = A056109(n) - 1. (End)
From Leo Tavares, Oct 06 2022: (Start)
a(n) = A003154(n+1) - A000567(n+1). See Split Stars illustration.
a(n) = A014105(n) + 2*A000217(n). (End)

A028896 6 times triangular numbers: a(n) = 3n(n+1).

Original entry on oeis.org

0, 6, 18, 36, 60, 90, 126, 168, 216, 270, 330, 396, 468, 546, 630, 720, 816, 918, 1026, 1140, 1260, 1386, 1518, 1656, 1800, 1950, 2106, 2268, 2436, 2610, 2790, 2976, 3168, 3366, 3570, 3780, 3996, 4218, 4446, 4680, 4920, 5166, 5418, 5676

Offset: 0

Joe Keane (jgk(AT)jgk.org), Dec 11 1999

From Floor van Lamoen, Jul 21 2001: (Start)
Write 1,2,3,4,... in a hexagonal spiral around 0; then a(n) is the sequence found by reading the line from 0 in the direction 0, 6, ...
The spiral begins:
                 85--84--83--82--81--80
                 /                     \
               86  56--55--54--53--52  79
               /   /                 \   \
             87  57  33--32--31--30  51  78
             /   /   /             \   \   \
           88  58  34  16--15--14  29  50  77
           /   /   /   /         \   \   \   \
         89  59  35  17   5---4  13  28  49  76
         /   /   /   /   /     \   \   \   \   \
    <==90==60==36==18===6===0   3  12  27  48  75
           /   /   /   /   /   /   /   /   /   /
         61  37  19   7   1---2  11  26  47  74
           \   \   \   \         /   /   /   /
           62  38  20   8---9--10  25  46  73
             \   \   \             /   /   /
             63  39  21--22--23--24  45  72
               \   \                 /   /
               64  40--41--42--43--44  71
                 \                     /
                 65--66--67--68--69--70
(End)
If Y is a 4-subset of an n-set X then, for n >= 5, a(n-5) is the number of (n-4)-subsets of X having exactly two elements in common with Y. - Milan Janjic, Dec 28 2007
a(n) is the maximal number of points of intersection of n+1 distinct triangles drawn in the plane. For example, two triangles can intersect in at most a(1) = 6 points (as illustrated in the Star of David configuration). - Terry Stickels (Terrystickels(AT)aol.com), Jul 12 2008
Also sequence found by reading the line from 0, in the direction 0, 6, ... and the same line from 0, in the direction 0, 18, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Axis perpendicular to A195143 in the same spiral. - Omar E. Pol, Sep 18 2011
Partial sums of A008588. - R. J. Mathar, Aug 28 2014
Also the number of 5-cycles in the (n+5)-triangular honeycomb acute knight graph. - Eric W. Weisstein, Jul 27 2017
a(n-4) is the maximum irregularity over all maximal 3-degenerate graphs with n vertices.  The extremal graphs are 3-stars (K_3 joined to n-3 independent vertices).  (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - Allan Bickle, May 29 2023

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Allan Bickle and Zhongyuan Che, Irregularities of Maximal k-degenerate Graphs, Discrete Applied Math. 331 (2023) 70-87.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Enrique Navarrete and Daniel Orellana, Finding Prime Numbers as Fixed Points of Sequences, arXiv:1907.10023 [math.NT], 2019.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Leo Tavares, Illustration: Centroid Hexagons.
Eric Weisstein's World of Mathematics, Graph Cycle.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000567, A003215, A008588, A024966, A028895, A033996, A046092, A049598, A084939, A084940, A084941, A084942, A084943, A084944, A124080.
Cf. A002378 (3-cycles in triangular honeycomb acute knight graph), A045943 (4-cycles), A152773 (6-cycles).
Cf. A007531.
The partial sums give A007531. - Leo Tavares, Jan 22 2022
Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).

List([0..44],n->3*n*(n+1)); # Muniru A Asiru, Mar 15 2019

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

[seq(6*binomial(n,2),n=1..44)]; # Zerinvary Lajos, Nov 24 2006

6 Accumulate[Range[0, 50]] (* Harvey P. Dale, Mar 05 2012 *)
6 PolygonalNumber[Range[0, 20]] (* Eric W. Weisstein, Jul 27 2017 *)
LinearRecurrence[{3, -3, 1}, {0, 6, 18}, 20] (* Eric W. Weisstein, Jul 27 2017 *)

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

first(n) = Vec(6*x/(1 - x)^3 + O(x^n), -n) \\ Iain Fox, Feb 14 2018

def A028896(n): return 3*n*(n+1) # Chai Wah Wu, Aug 07 2025

O.g.f.: 6*x/(1 - x)^3.
E.g.f.: 3*x*(x + 2)*exp(x). - G. C. Greubel, Aug 19 2017
a(n) = 6*A000217(n).
a(n) = polygorial(3, n+1). - Daniel Dockery (peritus(AT)gmail.com), Jun 16 2003
From Zerinvary Lajos, Mar 06 2007: (Start)
a(n) = A049598(n)/2.
a(n) = A124080(n) - A046092(n).
a(n) = A033996(n) - A002378(n). (End)
a(n) = A002378(n)*3 = A045943(n)*2. - Omar E. Pol, Dec 12 2008
a(n) = a(n-1) + 6*n for n>0, a(0)=0. - Vincenzo Librandi, Aug 05 2010
a(n) = A003215(n) - 1. - Omar E. Pol, Oct 03 2011
From Philippe Deléham, Mar 26 2013: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0)=0, a(1)=6, a(2)=18.
a(n) = A174709(6*n + 5). (End)
a(n) = A049450(n) + 4*n. - Lear Young, Apr 24 2014
a(n) = Sum_{i = n..2*n} 2*i. - Bruno Berselli, Feb 14 2018
a(n) = A320047(1, n, 1). - Kolosov Petro, Oct 04 2018
a(n) = T(3*n) - T(2*n-2) + T(n-2), where T(n) = A000217(n). In general, T(k)*T(n) = Sum_{i=0..k-1} (-1)^i*T((k-i)*(n-i)). - Charlie Marion, Dec 04 2020
From Amiram Eldar, Feb 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 1/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/3 - 1/3. (End)
From Amiram Eldar, Feb 21 2023: (Start)
Product_{n>=1} (1 - 1/a(n)) = -(3/Pi)*cos(sqrt(7/3)*Pi/2).
Product_{n>=1} (1 + 1/a(n)) = (3/Pi)*cosh(Pi/(2*sqrt(3))). (End)

A049451 Twice second pentagonal numbers.

Original entry on oeis.org

0, 4, 14, 30, 52, 80, 114, 154, 200, 252, 310, 374, 444, 520, 602, 690, 784, 884, 990, 1102, 1220, 1344, 1474, 1610, 1752, 1900, 2054, 2214, 2380, 2552, 2730, 2914, 3104, 3300, 3502, 3710, 3924, 4144, 4370, 4602, 4840, 5084, 5334, 5590, 5852, 6120, 6394, 6674, 6960, 7252, 7550, 7854

Offset: 0

Joe Keane (jgk(AT)jgk.org)

From Floor van Lamoen, Jul 21 2001: (Start)
Write 1,2,3,4,... in a hexagonal spiral around 0, then a(n) is the sequence found by reading the line from 0 in the direction 0,4,... . The spiral begins:
                            .
                          52
                          . \
            33--32--31--30  51
            /           . \   \
          34  16--15--14  29  50
          /   /       . \   \   \
        35  17   5---4  13  28  49
        /   /   /   . \   \   \   \
      36  18   6   0   3  12  27  48
      /   /   /   /   /   /   /   /
    37  19   7   1---2  11  26  47
      \   \   \         /   /   /
      38  20   8---9--10  25  46
        \   \             /   /
        39  21--22--23--24  45
          \                 /
          40--41--42--43--44
(End)
Number of edges in the join of the complete bipartite graph of order 2n and the cycle graph of order n, K_n,n * C_n. - Roberto E. Martinez II, Jan 07 2002
The average of the first n elements starting from a(1) is equal to (n+1)^2. - Mario Catalani (mario.catalani(AT)unito.it), Apr 10 2003
If Y is a 4-subset of an n-set X then, for n >= 4, a(n-4) is the number of (n-4)-subsets of X having either one element or two elements in common with Y. - Milan Janjic, Dec 28 2007
With offset 1: the maximum possible sum of numbers in an N x N standard Minesweeper grid. - Dmitry Kamenetsky, Dec 14 2008
a(n) = A001399(6*n-2), number of partitions of 6*n-2 into parts < 4. For example a(2)=14 where the partitions of 6*2-2=10 into parts < 4 are [1,1,1,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,2], [1,1,1,1,1,1,1,3], [1,1,1,1,1,1,2,2], [1,1,1,1,1,2,3], [1,1,1,1,2,2,2], [1,1,1,1,3,3], [1,1,1,2,2,3], [1,1,2,2,2,2], [1,1,2,3,3], [1,2,2,2,3], [2,2,2,2,2], [1,3,3,3], [2,2,3,3]. - Adi Dani, Jun 07 2011
A003056 is the following array A read by antidiagonals:
  0,  1,  2,  3,  4,  5, ...
  1,  2,  3,  4,  5,  6, ...
  2,  3,  4,  5,  6,  7, ...
  3,  4,  5,  6,  7,  8, ...
  4,  5,  6,  7,  8,  9, ...
  5,  6,  7,  8,  9, 10, ...
and a(n) is the hook sum Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). - R. J. Mathar, Jun 30 2013
a(n)*Pi is the total length of 3 points circle center spiral after n rotations. The spiral length at each rotation (L(n)) is A016957. The spiral length ratio rounded down [floor(L(n)/L(1))] is A001651. See illustration in links. - Kival Ngaokrajang, Dec 27 2013
Partial sums give A114364. - Leo Tavares, Feb 25 2022
For n >= 1, the continued fraction expansion of sqrt(27*a(n)) is [9n+1; {2, 2n-1, 1, 4, 1, 2n-1, 2, 18n+2}]. - Magus K. Chu, Oct 13 2022

			From _Dmitry Kamenetsky_, Dec 14 2008, with slight rewording by Raymond Martineau (mart0258(AT)yahoo.com), Dec 16 2008: (Start)
For an N x N Minesweeper grid the highest sum of numbers is (N-1)(3*N-2). This is achieved by filling every second row with mines (shown as 'X'). For example, when N=5 the best grids are:
.
  X X X X X
  4 6 6 6 4
  X X X X X
  4 6 6 6 4
  X X X X X
.
  and
.
  2 3 3 3 2
  X X X X X
  4 6 6 6 4
  X X X X X
  2 3 3 3 2
.
each giving a total of 52. (End)

L. B. W. Jolley, Summation of Series, Dover Publications, 1961, p. 12.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Raghavendra N. Bhat, Cristian Cobeli, and Alexandru Zaharescu, A lozenge triangulation of the plane with integers, arXiv:2403.10500 [math.NT], 2024.
Kival Ngaokrajang, Illustration of 3 points circle center spiral.
Leo Tavares, Illustration: Double Hexagonal Trapezoids.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000567, A001105, A002378, A005449, A033580, A049450.
Similar sequences are listed in A316466.
Cf. A003215, A005408, A114364.

List([0..55], n-> n*(3*n+1)); # G. C. Greubel, Sep 01 2019

a049451 n = n * (3 * n + 1)  -- Reinhard Zumkeller, Jul 07 2012

[n*(3*n+1): n in [0..55]]; // G. C. Greubel, Sep 01 2019

Table[n(3n+1), {n,0,55}] (* or *) CoefficientList[Series[2x(2+x)/(1-x)^3, {x,0,55}], x] (* Michael De Vlieger, Apr 05 2017 *)

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

[n*(3*n+1) for n in range(60)] # Gennady Eremin, Feb 27 2022

[n*(3*n+1) for n in (0..55)] # G. C. Greubel, Sep 01 2019

a(n) = n*(3*n+1).
G.f.: 2*x*(2+x)/(1-x)^3.
Sum_{i=1..n} a(i) = A045991(n+1). - Gary W. Adamson, Dec 20 2006
a(n) = 2*A005449(n). - Omar E. Pol, Dec 18 2008
a(n) = a(n-1) + 6*n -2, n > 0. - Vincenzo Librandi, Aug 06 2010
a(n) = A100104(n+1) - A100104(n). - Reinhard Zumkeller, Jul 07 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0) = 0, a(1) = 4, a(2) = 14. - Philippe Deléham, Mar 26 2013
a(n) = A174709(6*n+3). - Philippe Deléham, Mar 26 2013
a(n) = (24/(n+2)!)*Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*j^(n+2). - Bruno Berselli, Jun 04 2013 - after the similar formula of Vladimir Kruchinin in A002411
a(n) = A002061(n+1) + A056220(n). - Bruce J. Nicholson, Sep 21 2017
a(n) = Sum_{i = 2..5} P(i,n), where P(i,m) = m*((i-2)*m-(i-4))/2. - Bruno Berselli, Jul 04 2018
E.g.f.: x*(4 + 3*x)*exp(x). - G. C. Greubel, Sep 01 2019
a(n) = A003215(n) - A005408(n). - Leo Tavares, Feb 25 2022
From Amiram Eldar, Feb 27 2022: (Start)
Sum_{n>=1} 1/a(n) = 3 - Pi/(2*sqrt(3)) - 3*log(3)/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/sqrt(3) + 2*log(2) - 3. (End)
a(n) = A001105(n) + A002378(n). - Torlach Rush, Jul 11 2022

A131242 Partial sums of A059995: a(n) = sum_{k=0..n} floor(k/10).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 156, 162, 168, 174, 180, 186, 192, 198

Offset: 0

Hieronymus Fischer, Jun 21 2007

Complementary with A130488 regarding triangular numbers, in that A130488(n)+10*a(n)=n(n+1)/2=A000217(n).

			As square array :
    0,   0,   0,   0,   0,   0,   0,   0,   0,    0
    1,   2,   3,   4,   5,   6,   7,   8,   9,   10
   12,  14,  16,  18,  20,  22,  24,  26,  28,   30
   33,  36,  39,  42,  45,  48,  51,  54,  57,   60
   64,  68,  72,  76,  80,  84,  88,  92,  96,  100
  105, 110, 115, 120, 125, 130, 135, 140, 145,  150
  156, 162, 168, 174, 180, 186, 192, 198, 204,  210
... - _Philippe Deléham_, Mar 27 2013

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,1,-2,1).

Cf. A008728, A059995, A010879, A002266, A130488, A000217, A002620, A130518, A130519, A130520, A174709, A174738, A118729, A218470.

Table[(1/2)*Floor[n/10]*(2*n - 8 - 10*Floor[n/10]), {n,0,50}] (* G. C. Greubel, Dec 13 2016 *)
Accumulate[Table[FromDigits[Most[IntegerDigits[n]]],{n,0,110}]] (* or *) LinearRecurrence[{2,-1,0,0,0,0,0,0,0,1,-2,1},{0,0,0,0,0,0,0,0,0,0,1,2},120] (* Harvey P. Dale, Apr 06 2017 *)

for(n=0,50, print1((1/2)*floor(n/10)*(2n-8-10*floor(n/10)), ", ")) \\ G. C. Greubel, Dec 13 2016

a(n)=my(k=n\10); k*(n-5*k-4) \\ Charles R Greathouse IV, Dec 13 2016

a(n) = (1/2)*floor(n/10)*(2n-8-10*floor(n/10)).
a(n) = A059995(n)*(2n-8-10*A059995(n))/2.
a(n) = (1/2)*A059995(n)*(n-8+A010879(n)).
a(n) = (n-A010879(n))*(n+A010879(n)-8)/20.
G.f.: x^10/((1-x^10)(1-x)^2).
From Philippe Deléham, Mar 27 2013: (Start)
a(10n)   = A051624(n).
a(10n+1) = A135706(n).
a(10n+2) = A147874(n+1).
a(10n+3) = 2*A005476(n).
a(10n+4) = A033429(n).
a(10n+5) = A202803(n).
a(10n+6) = A168668(n).
a(10n+7) = 2*A147875(n).
a(10n+8) = A135705(n).
a(10n+9) = A124080(n). (End)
a(n) = A008728(n-10) for n>= 10. - Georg Fischer, Nov 03 2018

A174738 Partial sums of floor(n/7).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 17, 19, 21, 24, 27, 30, 33, 36, 39, 42, 46, 50, 54, 58, 62, 66, 70, 75, 80, 85, 90, 95, 100, 105, 111, 117, 123, 129, 135, 141, 147, 154, 161, 168, 175, 182, 189, 196, 204, 212, 220, 228, 236

Offset: 0

Mircea Merca, Nov 30 2010

Apart from the initial zeros, the same as A011867.

			a(9) = floor(0/7) + floor(1/7) + floor(2/7) + floor(3/7) + floor(4/7) + floor(5/7) + floor(6/7) + floor(7/7) + floor(8/7) + floor(9/7) = 3.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions, J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,1,-2,1).

Cf. A001106, A022264, A022265, A024966, A179986, A186029, A218471.

Cf. Similar sequences: A000217, A002620, A130518, A130519, A130520, A174709, A118729, A218470, A131242.

List([0..60], n-> Int((n-2)*(n-3)/14)); # G. C. Greubel, Aug 31 2019

[Round(n*(n-5)/14): n in [0..60]]; // Vincenzo Librandi, Jun 22 2011

A174738 := proc(n) round(n*(n-5)/14) ; end proc:
seq(A174738(n),n=0..30) ;

Table[Floor[(n - 2)*(n - 3)/14], {n,0,60}] (* G. C. Greubel, Dec 13 2016 *)

a(n)=(n-2)*(n-3)\14 \\ Charles R Greathouse IV, Sep 24 2015

[floor((n-2)*(n-3)/14) for n in (0..60)] # G. C. Greubel, Aug 31 2019

a(n) = round(n*(n-5)/14).
a(n) = floor((n-2)*(n-3)/14).
a(n) = ceiling((n+1)*(n-6)/14).
a(n) = a(n-7) + n - 6, n > 6.
a(n) = +2*a(n-1) - a(n-2) + a(n-7) - 2*a(n-8) + a(n-9). - R. J. Mathar, Nov 30 2010
G.f.: x^7/( (1 + x + x^2 + x^3 + x^4 + x^5 + x^6)*(1-x)^3 ). - R. J. Mathar, Nov 30 2010
a(7n) = A001106(n), a(7n+1) = A218471(n), a(7n+2) = A022264(n), a(7n+3) = A022265(n), a(7n+4) = A186029(n), a(7n+5) = A179986(n), a(7n+6) = A024966(n). - Philippe Deléham, Mar 26 2013

A118729 Rectangular array where row r contains the 8 numbers 4r^2 - 3r, 4r^2 - 2r, ..., 4r^2 + 4r.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 27, 30, 33, 36, 39, 42, 45, 48, 52, 56, 60, 64, 68, 72, 76, 80, 85, 90, 95, 100, 105, 110, 115, 120, 126, 132, 138, 144, 150, 156, 162, 168

Offset: 0

Stuart M. Ellerstein (ellerstein(AT)aol.com), May 21 2006

The numbers in row r span the interval ]8*A000217(r-1), 8*A000217(r)].
The first difference between the entries in row r is r.
Partial sums of floor(n/8). - Philippe Deléham, Mar 26 2013
Apart from the initial zeros, the same as A008726. - Philippe Deléham, Mar 28 2013
a(n+7) is the number of key presses required to type a word of n letters, all different, on a keypad with 8 keys where 1 press of a key is some letter, 2 presses is some other letter, etc., and under an optimal mapping of letters to keys and presses (answering LeetCode problem 3014). - Christopher J. Thomas, Feb 16 2024

			The array starts, with row r=0, as
  r=0:   0  0  0  0  0  0  0  0;
  r=1:   1  2  3  4  5  6  7  8;
  r=2:  10 12 14 16 18 20 22 24;
  r=3:  27 30 33 36 39 42 45 48;

G. C. Greubel, Table of n, a(n) for n = 0..1000
LeetCode, 3014. Minimum Number of Pushes to Type Word I.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,1,-2,1).

Cf. similar sequences: A000217, A002620, A130518, A130519, A130520, A174709, A174738, A218470, A131242.

Flatten[Table[4r^2+r(Range[-3,4]),{r,0,6}]] (* or *) LinearRecurrence[ {2,-1,0,0,0,0,0,1,-2,1},{0,0,0,0,0,0,0,0,1,2},60] (* Harvey P. Dale, Nov 26 2015 *)

From Philippe Deléham, Mar 26 2013: (Start)
a(8k)   = A001107(k).
a(8k+1) = A002939(k).
a(8k+2) = A033991(k).
a(8k+3) = A016742(k).
a(8k+4) = A007742(k).
a(8k+5) = A002943(k).
a(8k+6) = A033954(k).
a(8k+7) = A033996(k). (End)
G.f.: x^8/((1-x)^2*(1-x^8)). - Philippe Deléham, Mar 28 2013
a(n) = floor(n/8)*(n-3-4*floor(n/8)). - Ridouane Oudra, Jun 04 2019
a(n+7) = (1/2)*(n+(n mod 8))*(floor(n/8)+1). - Christopher J. Thomas, Feb 13 2024

Redefined as a rectangular tabf array and description simplified by R. J. Mathar, Oct 20 2010

A008724 a(n) = floor(n^2/12).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 18, 21, 24, 27, 30, 33, 36, 40, 44, 48, 52, 56, 60, 65, 70, 75, 80, 85, 90, 96, 102, 108, 114, 120, 126, 133, 140, 147, 154, 161, 168, 176, 184, 192, 200, 208, 216, 225, 234, 243, 252, 261, 270, 280, 290, 300, 310, 320, 330, 341, 352

Offset: 0

N. J. A. Sloane

With a different offset, Molien series for 3-dimensional group [2,n] = *22n.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
P. T. Ho, The crossing number of K_{4,n} on the real projective plane, Discr. Math., 304 (2005), pp. 23-33.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 189.
Eric Weisstein's World of Mathematics, Toroidal Crossing Number.
Index entries for Molien series.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,1,-2,1).

Cf. A001399, A174709.

List([0..70], n-> Int(n^2/12) ); # G. C. Greubel, Sep 09 2019

a008724:=func< n | Floor(n^2/12) >; [ a008724(n): n in [0..70] ];

A008724 := proc(n)
    floor(n^2/12) ;
end proc:
seq(A008724(n),n=0..30) ; # R. J. Mathar, Mar 28 2017

Floor[Range[0, 70]^2/12] (* G. C. Greubel, Sep 09 2019 *)

a(n)=n^2\12 \\ Charles R Greathouse IV, Jul 02 2013

[floor(n^2/12) for n in (0..70)] # G. C. Greubel, Sep 09 2019

a(n) = a(n-6) + n - 3. - Paul Barry, Jul 14 2004
a(n) = Sum_{j=0..n+2} floor(j/6), a(n-2) = (1/2)*floor(n/6)*(2*n - 4 - 6*floor(n/6)). - Mitch Harris, Sep 08 2008
G.f.: x^4/((1-x)^2*(1-x^6)).
Sum_{n>=4} 1/a(n) = Pi^2/18 - Pi/(2*sqrt(3)) + 49/12. - Amiram Eldar, Aug 14 2022
a(n) = a(-n) = A174709(n+2). - Michael Somos, Dec 05 2023

Minor edits by Klaus Brockhaus, Nov 24 2010

cp's OEIS Frontend

A000567 Octagonal numbers: n*(3*n-2). Also called star numbers.

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A033428 a(n) = 3*n^2.

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A045944 Rhombic matchstick numbers: a(n) = n*(3*n+2).

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A028896 6 times triangular numbers: a(n) = 3*n*(n+1).

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A049451 Twice second pentagonal numbers.

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A000567 Octagonal numbers: n(3n-2). Also called star numbers.

A045944 Rhombic matchstick numbers: a(n) = n(3n+2).

A028896 6 times triangular numbers: a(n) = 3n(n+1).

A118729 Rectangular array where row r contains the 8 numbers 4r^2 - 3r, 4r^2 - 2r, ..., 4r^2 + 4r.