A045946 - OEIS

%I #36 Jun 12 2021 02:44:19

%S 0,24,84,180,312,480,684,924,1200,1512,1860,2244,2664,3120,3612,4140,

%T 4704,5304,5940,6612,7320,8064,8844,9660,10512,11400,12324,13284,

%U 14280,15312,16380,17484,18624,19800,21012,22260,23544,24864,26220,27612

%N Star of David matchstick numbers: 6*n*(3*n+1).

%C Vertical spoke of triangular spiral in A051682. - _Paul Barry_, Mar 15 2003

%H Ivan Panchenko, <a href="/A045946/b045946.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = 24*C(n, 1) + 36*C(n, 2); binomial transform of (0, 24, 36, 0, 0, 0, ...). - _Paul Barry_, Mar 15 2003

%F 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

%F G.f.: 12*x*(2+x)/(1-x)^3. - _Ivan Panchenko_, Nov 13 2013

%F a(n) = 2*A045945(n). - _Michel Marcus_, Nov 13 2013

%F a(n) = 12*A005449(n). - _R. J. Mathar_, Feb 08 2016

%F From _Amiram Eldar_, Jan 14 2021: (Start)

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

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

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

%o (PARI) a(n)=18*n^2+6*n \\ _Charles R Greathouse IV_, Feb 19 2017

%Y Cf. A005449, A045945, A051682.

%K nonn,nice,easy

%O 0,2

%A _R. K. Guy_