login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A045944 Rhombic matchstick numbers: n*(3*n+2). 57
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 (list; graph; refs; listen; history; text; internal format)
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 0,5,.... 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: 4T_n + S_n where T_n is triangular number n and S_n is square number n. - 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

LINKS

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.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

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(6n+4). - Philippe Deléham, Mar 26 2013

a(n) = A001082(2n). - 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

MATHEMATICA

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 *)

PROG

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

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

CROSSREFS

Bisection of A001859. See Comments of A135713.

Cf. A000567, A049450.

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.

Sequence in context: A063076 A270805 A132479 * A038361 A227719 A172166

Adjacent sequences:  A045941 A045942 A045943 * A045945 A045946 A045947

KEYWORD

nonn,easy,nice,changed

AUTHOR

R. K. Guy

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified September 26 08:27 EDT 2017. Contains 292506 sequences.