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

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

Offset

0,2

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).

Formula

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

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)

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 *)

Prog

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

Crossrefs

Cf. A006516, A016152.

Sequence in context: A374513 A377200 A125402 * A003457 A263822 A016241

Adjacent sequences: A016287 A016288 A016289 * A016291 A016292 A016293

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Status

approved