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 0,2,.... The spiral begins:
.
56--55--54--53--52
/ \
57 33--32--31--30 51
/ / \ \
58 34 16--15--14 29 50
/ / / \ \ \
59 35 17 5---4 13 28 49
/ / / / \ \ \ \
60 36 18 6 0 3 12 27 48
/ / / / / . / / / /
61 37 19 7 1---2 11 26 47
\ \ \ \ . / / /
62 38 20 8---9--10 25 46
\ \ \ . / /
63 39 21--22--23--24 45
\ \ . /
64 40--41--42--43--44
\ .
65--66--67--68--69--70
(End)
Starting with offset 1 = binomial transform of [2, 8, 6, 0, 0, 0, ...]. - Gary W. Adamson, Jan 09 2009
Number of possible pawn moves on an (n+1) X (n+1) chessboard (n=>3). - Johannes W. Meijer, Feb 04 2010
Even octagonal numbers divided by 4. - Omar E. Pol, Aug 19 2011
Partial sums give A011379. - Omar E. Pol, Jan 12 2013
First differences are A016933; second differences equal 6. - Bob Selcoe, Apr 02 2015
For n >= 1, the continued fraction expansion of sqrt(27*a(n)) is [9n-2; {2, 2n-1, 6, 2n-1, 2, 18n-4}]. - Magus K. Chu, Oct 13 2022
LINKS
Ivan Panchenko, Table of n, a(n) for n = 0..1000
Richard P. Brent, Generalising Tuenter's binomial sums, arXiv:1407.3533 [math.CO], 2014. (page 16)
Leo Tavares, Illustration: X Hexagons
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
O.g.f.: A(x) = 2*x*(1+2*x)/(1-x)^3.
a(n) = 2*A000326(n), twice pentagonal numbers. - Omar E. Pol, May 14 2008
a(n) = a(n-1) + 6*n - 4 (with a(0)=0). - Vincenzo Librandi, Aug 06 2010
E.g.f.: x*(2 + 3*x)*exp(x). - Vincenzo Librandi, Apr 28 2016
a(n) = (2/3)*A000217(3*n-1). - Bruno Berselli, Feb 13 2017
From Amiram Eldar, Feb 20 2022: (Start)
Sum_{n>=1} 1/a(n) = 3*log(3)/2 - Pi/(2*sqrt(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/sqrt(3) - 2*log(2). (End)
From Leo Tavares, Feb 23 2022: (Start)
EXAMPLE
On a 4 X 4 chessboard pawns at the second row have (3+4+4+3) moves and pawns at the third row have (2+3+3+2) moves so a(3) = 24. - Johannes W. Meijer, Feb 04 2010
From Adi Dani, Jun 04 2011: (Start)
a(1)=2: the partitions of 6*1-1=5 into 3 parts are [1,1,3] and[1,2,2].
a(2)=10: the partitions of 6*2-1=11 into 3 parts are [1,1,9], [1,2,8], [1,3,7], [1,4,6], [1,5,5], [2,2,7], [2,3,6], [2,4,5], [3,3,5], and [3,4,4].
(End)
.
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. 2 10 24 44 70
- Philippe Deléham, Mar 30 2013
MAPLE
seq(n*(3*n-1), n=0..44); # Zerinvary Lajos, Jun 12 2007
MATHEMATICA
Table[n(3n-1), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 2, 10}, 50] (* Harvey P. Dale, Jun 21 2014 *)
2*PolygonalNumber[5, Range[0, 50]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 01 2018 *)
PROG
(PARI) a(n)=n*(3*n-1) \\ Charles R Greathouse IV, Nov 20 2012
(Magma) [n*(3*n-1) : n in [0..50]]; // Wesley Ivan Hurt, Sep 24 2017
(Sage) [n*(3*n-1) for n in (0..50)] # G. C. Greubel, Aug 31 2019
(GAP) List([0..50], n-> n*(3*n-1)); # G. C. Greubel, Aug 31 2019
CROSSREFS
Cf. A000567.
Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488. [Bruno Berselli, Jun 10 2013]
Cf. sequences listed in A254963.
KEYWORD
nonn,easy,nice
AUTHOR
Joe Keane (jgk(AT)jgk.org).
STATUS
approved