A016290 Expansion of g.f. 1/((1-2*x)*(1-4*x)*(1-8*x)).

1, 14, 140, 1240, 10416, 85344, 690880, 5559680, 44608256, 357389824, 2861214720, 22898104320, 183218384896, 1465881288704, 11727587164160, 93822844764160, 750591347982336, 6004765143465984, 48038258586419200, 384306618446643200, 3074455146595352576, 24595649968853745664

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

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

Links

• T. D. Noe, Table of n, a(n) for n=0..100
• Julia Crager, Felicia Flores, Timothy E. Goldberg, Lauren L. Rose, Daniel Rose-Levine, Darrion Thornburgh, and Raphael Walker, How many cards should you lay out in a game of EvenQuads? A detailed study of 2-caps in AG(n,2), arXiv:2212.05353 [math.CO], 2023.
• Index entries for linear recurrences with constant coefficients, signature (14,-56,64).

Crossrefs

Cf. A006516, A016152, A160873.

Programs

Magma

[(2^n-6*4^n+8*8^n)/3 : n in [0..20]]; // Wesley Ivan Hurt, Jul 07 2014

Maple

[seq(GBC(n+1,3,2)-GBC(n,3,2), n=2..30)]; # produces A016290 (cf. A006516).
seq((2^n-6*4^n+8*8^n)/3, n=0..20);
seq(binomial(2^n,3)/4, n=2..20); # Zerinvary Lajos, Feb 22 2008

Mathematica

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 *)

Formula

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

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

a(n) = (2^n - 6*4^n + 8*8^n)/3. - James R. Buddenhagen, Dec 14 2003

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

From Vincenzo Librandi, Mar 15 2011: (Start)

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

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

From Elmo R. Oliveira, Mar 26 2025: (Start)

E.g.f.: exp(2*x)*(8*exp(6*x) - 6*exp(2*x) + 1)/3.

a(n) = A160873(n+2)/3. (End)