A003674 -id:A003674 - cp's OEIS frontend

A003683 a(n) = 2^(n-1)*(2^n - (-1)^n)/3.

0, 1, 2, 12, 40, 176, 672, 2752, 10880, 43776, 174592, 699392, 2795520, 11186176, 44736512, 178962432, 715816960, 2863333376, 11453202432, 45813071872, 183251763200, 733008101376, 2932030308352, 11728125427712

Offset: 0

N. J. A. Sloane

a(n) = A001045(n) * A011782(n). - Paul Barry, May 20 2003
The sequence 1,2,12,... is the binomial transform of (1, 1, 9, 9, 81, 81, ...) = 2*3^n/3 + (-3)^n/3. - Paul Barry, Jul 17 2003
Form a graph whose adjacency matrix is the tensor product of that of C_3 and [1,1;1,1]. a(n) counts walks of length n between any pair of adjacent nodes. A054881(n) counts closed walks of length n at a node.
Arises in connection with merit factor of the GRS sequences - see Hoeholdt et al.
2*a(n) = the constant term of the reduction by x^2->x+2 of the polynomial p(n,x) = ((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+2); see A192382.  For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232. - Clark Kimberling, Jun 30 2011
Apparently a(n+1) is the number of 3D tilings of a 2 X 2 X n room with bricks of 1 X 2 X 2 shape. - R. J. Mathar, Dec 06 2013
The ratio a(n+1)/a(n) converges to 4 as n approaches infinity. - Felix P. Muga II, Mar 10 2014

M. Gardner, Riddles of the Sphinx, New Mathematical Library, M.A.A., 1987, p. 145. Math. Rev. 89i:00015.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
T. Hoeholdt, H. E. Jensen, and J. Justesen, Aperiodic correlations and the merit factor of a class of binary sequences, IEEE Trans. Inform. Theory, 13 (1985), 549-552
R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 7.
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
Eric Weisstein's World of Mathematics, Octahedral Graph
Index entries for linear recurrences with constant coefficients, signature (2,8).

Cf. A001045, A003674, A011782, A054881, A192232, A192382, A238801.

[2^(n-1)*(2^n - (-1)^n)/3: n in [0..30]]; // Vincenzo Librandi, Aug 19 2011

A003683:=n->2^(n-1)*(2^n - (-1)^n)/3; seq(A003683(n), n=0..50); # Wesley Ivan Hurt, Dec 06 2013

Table[2^(n-1) (2^n-(-1)^n)/3,{n,0,30}] (* or *) LinearRecurrence[{2,8},{0,1},30] (* Harvey P. Dale, Sep 15 2013 *)

PARI
```
a(n)=if(n<0,0,2^(n-1)*(2^n-(-1)^n)/3)
```

a(n)=(2^n-(-1)^n)<<(n-1)/3 \\ Charles R Greathouse IV, Apr 17 2012

[lucas_number1(n,2,-8) for n in range(0, 24)] # Zerinvary Lajos, Apr 22 2009

a(n) = A003674(n)/3.
a(n) = 2*a(n-1) + 8*a(n-2), with a(0)=0, a(1)=1. - Barry E. Williams, Jan 04 2000
G.f.: x/((1+2*x)*(1-4*x)).
a(n) = ((1+3)^n-(1-3)^n)/6. - Paul Barry, May 14 2003
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k+1)*9^k. - Paul Barry, May 20 2003
E.g.f.: exp(x)*sinh(3*x)/3. - Paul Barry, Jul 09 2003
a(n+1) = 2^n*A001045(n+1). - R. J. Mathar, Jul 08 2009
a(n+1) = Sum_{k=0..n} A238801(n,k)*3^k. - Philippe Deléham, Mar 07 2014

Erroneous references to spanning trees in K_2 X P_n deleted by Frans Faase, Feb 07 2009

A071930 Number of words of length 2n in the two letters s and t that reduce to the identity 1 by using the relations ssTT=1, ststSS=1 and ststTT=1, where S and T are the inverses of s and t, respectively (i.e., sS=1 and tT=1). The generators s and t and the three stated relations generate the quaternion group Q4.

Original entry on oeis.org

0, 6, 12, 72, 240, 1056, 4032, 16512, 65280, 262656, 1047552, 4196352, 16773120, 67117056, 268419072, 1073774592, 4294901760, 17180000256, 68719214592, 274878431232, 1099510579200, 4398048608256, 17592181850112

Offset: 1

John W. Layman and Jamaine Paddyfoot (jay_paddyfoot(AT)hotmail.com), Jun 14 2002

a(n) = A003683(n+1)/6. No words of odd length (see the description above) reduce to 1.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,8).

Cf. A003674, A003683.

[4^(n-1)-(-2)^(n-1): n in [1..40]]; // G. C. Greubel, Feb 17 2023

Table[2^(2n-2)-(-2)^(n-1),{n,30}] (* or *) LinearRecurrence[{2,8},{0,6},30] (* Harvey P. Dale, Dec 10 2012 *)

[4^(n-1)-(-2)^(n-1) for n in range(1,41)] # G. C. Greubel, Feb 17 2023

a(n) = 2^(2*n-2) - (-2)^(n-1) = 6*A003683(n-1).
From Harvey P. Dale, Dec 10 2012: (Start)
a(n) = 2*a(n-1) + 8*a-(n-2).
G.f.: 6*x^2/(1-2*x-8*x^2). (End)
G.f.: Q(0), where Q(k)= 1 - 1/(4^k - 4*x*16^k/(4*x*4^k - 1/(1 + 1/(2*4^k - 16*x*16^k/(8*x*4^k +1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 21 2013
a(n) = 2*A003674(n-1). - G. C. Greubel, Feb 17 2023

cp's OEIS Frontend

A003683 a(n) = 2^(n-1)*(2^n - (-1)^n)/3.

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