A049450 Pentagonal numbers multiplied by 2: a(n) = n*(3*n-1).

0, 2, 10, 24, 44, 70, 102, 140, 184, 234, 290, 352, 420, 494, 574, 660, 752, 850, 954, 1064, 1180, 1302, 1430, 1564, 1704, 1850, 2002, 2160, 2324, 2494, 2670, 2852, 3040, 3234, 3434, 3640, 3852, 4070, 4294, 4524, 4760, 5002, 5250, 5504, 5764

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

a(n) = A069905(6n-1): Number of partitions of 6*n-1 into 3 parts. - Adi Dani, Jun 04 2011

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) = A049452(n) - A033428(n). - Zerinvary Lajos, Jun 12 2007

a(n) = 2*A000326(n), twice pentagonal numbers. - Omar E. Pol, May 14 2008

a(n) = A022264(n) - A000217(n). - Reinhard Zumkeller, Oct 09 2008

a(n) = a(n-1) + 6*n - 4 (with a(0)=0). - Vincenzo Librandi, Aug 06 2010

a(n) = A014642(n)/4 = A033579(n)/2. - Omar E. Pol, Aug 19 2011

a(n) = A000290(n) + A000384(n) = A000217(n) + A000566(n). - Omar E. Pol, Jan 11 2013

a(n+1) = A014107(n+2) + A000290(n). - Philippe Deléham, Mar 30 2013

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

a(n) = A002061(n) + A056220(n). - Bruce J. Nicholson, Sep 21 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)

a(n) = A003215(n) - A016813(n).

a(n) = 2*A000290(n) + 2*A000217(n-1). (End)

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)
.
.                                                         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 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
.    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.

Bisection of A001859. Cf. A045944, A000326, A033579, A027599, A049451.

Cf. A033586 (King), A035005 (Queen), A035006 (Rook), A035008 (Knight) and A002492 (Bishop).

Cf. numbers of the form n*(n*k-k+4))/2 listed in A226488. [Bruno Berselli, Jun 10 2013]

Cf. sequences listed in A254963.

Cf. A003215, A016813, A000290, A000217.

Sequence in context: A330280 A293412 A224837 * A092906 A244383 A295053

Adjacent sequences: A049447 A049448 A049449 * A049451 A049452 A049453

Keyword

nonn,easy,nice

Author

Joe Keane (jgk(AT)jgk.org).

Status

approved