OFFSET
0,2
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
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).
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) = 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) = 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
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)
EXAMPLE
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)
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 *)
PROG
(Haskell) a049451 n = n * (3 * n + 1) -- Reinhard Zumkeller, Jul 07 2012
(PARI) a(n)=n*(3*n+1) \\ Charles R Greathouse IV, Sep 24 2015
(Magma) [n*(3*n+1): n in [0..55]]; // G. C. Greubel, Sep 01 2019
(Sage) [n*(3*n+1) for n in (0..55)] # G. C. Greubel, Sep 01 2019
(GAP) List([0..55], n-> n*(3*n+1)); # G. C. Greubel, Sep 01 2019
(Python) [n*(3*n+1) for n in range(60)] # Gennady Eremin, Feb 27 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Joe Keane (jgk(AT)jgk.org)
STATUS
approved