A016290 - OEIS

%I #58 Aug 02 2023 14:14:32

%S 1,14,140,1240,10416,85344,690880,5559680,44608256,357389824,

%T 2861214720,22898104320,183218384896,1465881288704,11727587164160,

%U 93822844764160,750591347982336,6004765143465984,48038258586419200,384306618446643200,3074455146595352576,24595649968853745664

%N Expansion of 1/((1-2x)(1-4x)(1-8x)).

%C a(n) is the number of quads in the EvenQuads-2^{n+2} deck. - _Tanya Khovanova_ and MIT PRIMES STEP senior group, Jul 02 2023

%H T. D. Noe, <a href="/A016290/b016290.txt">Table of n, a(n) for n=0..100</a>

%H Julia Crager, Felicia Flores, Timothy E. Goldberg, Lauren L. Rose, Daniel Rose-Levine, Darrion Thornburgh, and Raphael Walker, <a href="https://arxiv.org/abs/2212.05353">How many cards should you lay out in a game of EvenQuads? A detailed study of 2-caps in AG(n,2)</a>, arXiv:2212.05353 [math.CO], 2023.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (14,-56,64).

%F G.f.: 1/((1-2x)*(1-4x)*(1-8x)).

%F Difference of Gaussian binomial coefficients [ n+1, 3 ] - [ n, 3 ] (n >= 2).

%F a(n) = (2^n-6*4^n+8*8^n)/3. - _James R. Buddenhagen_, Dec 14 2003

%F a(n) = Sum_{0<=i,j,k,<=n; i+j+k=n} 2^i*4^j*8^k. - _Hieronymus Fischer_, Jun 25 2007

%F From _Vincenzo Librandi_, Mar 15 2011: (Start)

%F a(n) = 14*a(n-1) - 56*a(n-2) + 64*a(n-3) for n >= 3.

%F a(n) = 12*a(n-1) - 32*a(n-2) + 2^n with a(0)=1, a(1)=14. (End)

%p [seq(GBC(n+1,3,2)-GBC(n,3,2), n=2..30)]; # produces A016290 (cf. A006516).

%p seq((2^n-6*4^n+8*8^n)/3, n=0..20);

%p seq(binomial(2^n,3)/4, n=2..20); # _Zerinvary Lajos_, Feb 22 2008

%t CoefficientList[Series[1/((1-2x)(1-4x)(1-8x)),{x,0,30}],x] (* or *) LinearRecurrence[{14,-56,64},{1,14,140},30] (* _Harvey P. Dale_, Jul 23 2011 *)

%o (Magma) [(2^n-6*4^n+8*8^n)/3 : n in [0..20]]; // _Wesley Ivan Hurt_, Jul 07 2014

%Y Cf. A006516, A016152.

%K nonn,nice,easy

%O 0,2

%A _N. J. A. Sloane_