A160873 -id:A160873 - cp's OEIS frontend

A016152 a(n) = 4^(n-1)*(2^n-1).

0, 1, 12, 112, 960, 7936, 64512, 520192, 4177920, 33488896, 268173312, 2146435072, 17175674880, 137422176256, 1099444518912, 8795824586752, 70367670435840, 562945658454016, 4503582447501312, 36028728299487232

Offset: 0

N. J. A. Sloane

Numbers whose binary representation is the concatenation of n digits 1 and 2(n-1) digits 0, for n>0. (See A147816.) - Omar E. Pol, Nov 13 2008
a(n) is the number of lattices L in Z^n such that the quotient group Z^n / L is C_8. - Álvar Ibeas, Nov 29 2015
a(n) is a maximum number of intercalates in a Latin square of order 2^n (see A092237). - Eduard I. Vatutin, Apr 30 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..140
Eduard I. Vatutin, Example of Latin squares of order 2^n with maximum number of intercalates.
Index entries for linear recurrences with constant coefficients, signature (12,-32).

Second column of triangle A075499.
Cf. A019677, A147538, A147816.
Cf. A065442, A092237.

[4^(n-1)*(2^n-1): n in [0..40]]; // Vincenzo Librandi, Apr 26 2011

Table[4^(n - 1) (2^n - 1), {n, 0, 19}] (* Michael De Vlieger, Nov 30 2015 *)

a(n)=4^(n-1)*(2^n-1) \\ Charles R Greathouse IV, Oct 07 2015

my(x='x+O('x^30)); concat(0, Vec(x/((1-4*x)*(1-8*x)))) \\ Altug Alkan, Dec 04 2015

[lucas_number1(n,12,32) for n in range(0, 20)] # Zerinvary Lajos, Apr 27 2009

From Barry E. Williams, Jan 17 2000: (Start)
a(n) = ((8^(n+1)) - 4^(n+1))/4.
a(n) = 12a(n-1) - 32a(n-2), n>0; a(0)=1. (End)
a(n) = (4^(n-1))*Stirling2(n+1, 2), n>=0, with Stirling2(n, m)=A008277(n, m).
a(n) = -4^(n-1) + 2*8^(n-1).
E.g.f. for a(n+1), n>=0: d^2/dx^2((((exp(4*x)-1)/4)^2)/2!) = -exp(4*x) + 2*exp(8*x).
G.f.: x/((1-4*x)*(1-8*x)).
((6+sqrt4)^n - (6-sqrt4)^n)/4 in Fibonacci form. Offset 1. a(3)=112. - Al Hakanson (hawkuu(AT)gmail.com), Dec 31 2008
a(n) + A160873(n) + A006096(n) = A006096(n+2), for n > 2. - Álvar Ibeas, Nov 29 2015
Sum_{n>0} 1/a(n) = 4*E - 16/3, where E is the Erdős-Borwein constant (A065442). - Peter McNair, Dec 19 2022

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

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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

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).

Cf. A006516, A016152, A160873.

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

[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

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 *)

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)

cp's OEIS Frontend

A016152 a(n) = 4^(n-1)*(2^n-1).

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

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cp's OEIS Frontend

A016152 a(n) = 4^(n-1)*(2^n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Sage

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

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