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A045944 Rhombic matchstick numbers: n*(3*n+2). 53
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

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,... - Floor van Lamoen (fvlamoen(AT)hotmail.com), Jul 21 2001. The spiral begins:

......16..15..14

....17..5...4...13

..18..6...0...3...12

19..7...1...2...11..26

..20..8...9...10..25

....21..22..23..24

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

REFERENCES

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.

LINKS

Ivan Panchenko, Table of n, a(n) for n = 0..1000

M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550, 2013

Index entries for sequences related to 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(0)=0, a(1)=5, a(2)=16, a(n)=3*a(n-1)-3*a(n-2)+a(n-3). [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(A016969(i), i=0..n-1) 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

MATHEMATICA

Table[n*(3n+2), {n, 0, 60}] (* Harvey P. Dale, May 05 2011 *)

PROG

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

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: A222535 A063076 A132479 * A038361 A227719 A172166

Adjacent sequences:  A045941 A045942 A045943 * A045945 A045946 A045947

KEYWORD

nonn,easy,nice

AUTHOR

R. K. Guy

EXTENSIONS

Removed a zero in an A-number - R. J. Mathar, Nov 22 2009

STATUS

approved

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Last modified December 18 23:18 EST 2014. Contains 252175 sequences.