A049451 Twice second pentagonal numbers.

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

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

• nonn
• easy

Comments

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

Examples

			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)
		

References

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

Links

• 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).

Crossrefs

Cf. A000567, A001105, A002378, A005449, A033580, A049450.

Similar sequences are listed in A316466.

Cf. A003215, A005408, A114364.

Programs

GAP

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

Haskell

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

Magma

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

Mathematica

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 *)

PARI

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

Python

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

Sage

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

Formula

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