A045946 Star of David matchstick numbers: 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

Offset

0,2

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).

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)

Mathematica

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

Prog

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

Crossrefs

Cf. A005449, A045945, A051682.

Sequence in context: A063456 A265646 A304158 * A192838 A289155 A101861

Adjacent sequences: A045943 A045944 A045945 * A045947 A045948 A045949

Keyword

nonn,nice,easy

Author

R. K. Guy

Status

approved