A045946 Star of David matchstick numbers: a(n) = 6*n*(3*n+1).

0, 24, 84, 180, 312, 480, 684, 924, 1200, 1512, 1860, 2244, 2664, 3120, 3612, 4140, 4704, 5304, 5940, 6612, 7320, 8064, 8844, 9660, 10512, 11400, 12324, 13284, 14280, 15312, 16380, 17484, 18624, 19800, 21012, 22260, 23544, 24864, 26220, 27612, 29040, 30504, 32004

Offset: 0

Author

R. K. Guy

Keywords

• nonn
• nice
• easy

Comments

Vertical spoke of triangular spiral in A051682. - Paul Barry, Mar 15 2003

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A005449, A033580, A049451, A045945, A051682, A081266.

Programs

Mathematica

Table[6n(3n+1),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,24,84},40] (* Harvey P. Dale, Nov 23 2012 *)

PARI

a(n)=18*n^2+6*n \\ Charles R Greathouse IV, Feb 19 2017

Formula

a(n) = 24*C(n,1) + 36*C(n,2); binomial transform of (0, 24, 36, 0, 0, 0, ...). - Paul Barry, Mar 15 2003

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=24, a(2)=84. - Harvey P. Dale, Nov 23 2012

G.f.: 12*x*(2+x)/(1-x)^3. - Ivan Panchenko, Nov 13 2013

a(n) = 2*A045945(n). - Michel Marcus, Nov 13 2013

a(n) = 12*A005449(n). - R. J. Mathar, Feb 08 2016

From Amiram Eldar, Jan 14 2021: (Start)

Sum_{n>=1} 1/a(n) = 1/2 - Pi/(12*sqrt(3)) - log(3)/4.

Sum_{n>=1} (-1)^(n+1)/a(n) = -1/2 + Pi/(6*sqrt(3)) + log(2)/3. (End)

From Elmo R. Oliveira, Dec 12 2024: (Start)

E.g.f.: 6*exp(x)*x*(4 + 3*x).

a(n) = 6*A049451(n) = 4*A081266(n) = 3*A033580(n). (End)