cp's OEIS Frontend

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

%I A003683 #77 Feb 16 2025 08:32:27
%S A003683 0,1,2,12,40,176,672,2752,10880,43776,174592,699392,2795520,11186176,
%T A003683 44736512,178962432,715816960,2863333376,11453202432,45813071872,
%U A003683 183251763200,733008101376,2932030308352,11728125427712
%N A003683 a(n) = 2^(n-1)*(2^n - (-1)^n)/3.
%C A003683 a(n) = A001045(n) * A011782(n). - _Paul Barry_, May 20 2003
%C A003683 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
%C A003683 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.
%C A003683 Arises in connection with merit factor of the GRS sequences - see Hoeholdt et al.
%C A003683 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
%C A003683 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
%C A003683 The ratio a(n+1)/a(n) converges to 4 as n approaches infinity. - _Felix P. Muga II_, Mar 10 2014
%D A003683 M. Gardner, Riddles of the Sphinx, New Mathematical Library, M.A.A., 1987, p. 145. Math. Rev. 89i:00015.
%H A003683 Vincenzo Librandi, <a href="/A003683/b003683.txt">Table of n, a(n) for n = 0..1000</a>
%H A003683 T. Hoeholdt, H. E. Jensen, and J. Justesen, <a href="http://dx.doi.org/10.1109/TIT.1985.1057071">Aperiodic correlations and the merit factor of a class of binary sequences</a>, IEEE Trans. Inform. Theory, 13 (1985), 549-552
%H A003683 R. J. Mathar, <a href="/A102518/a102518.pdf">Counting Walks on Finite Graphs</a>, Nov 2020, Section 7.
%H A003683 F. P. Muga II, <a href="https://www.researchgate.net/publication/267327689">Extending the Golden Ratio and the Binet-de Moivre Formula</a>, March 2014; Preprint on ResearchGate.
%H A003683 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OctahedralGraph.html">Octahedral Graph</a>
%H A003683 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,8).
%F A003683 a(n) = A003674(n)/3.
%F A003683 a(n) = 2*a(n-1) + 8*a(n-2), with a(0)=0, a(1)=1. - _Barry E. Williams_, Jan 04 2000
%F A003683 G.f.: x/((1+2*x)*(1-4*x)).
%F A003683 a(n) = ((1+3)^n-(1-3)^n)/6. - _Paul Barry_, May 14 2003
%F A003683 a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k+1)*9^k. - _Paul Barry_, May 20 2003
%F A003683 E.g.f.: exp(x)*sinh(3*x)/3. - _Paul Barry_, Jul 09 2003
%F A003683 a(n+1) = 2^n*A001045(n+1). - _R. J. Mathar_, Jul 08 2009
%F A003683 a(n+1) = Sum_{k=0..n} A238801(n,k)*3^k. - _Philippe Deléham_, Mar 07 2014
%p A003683 A003683:=n->2^(n-1)*(2^n - (-1)^n)/3; seq(A003683(n), n=0..50); # _Wesley Ivan Hurt_, Dec 06 2013
%t A003683 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 *)
%o A003683 (PARI) a(n)=if(n<0,0,2^(n-1)*(2^n-(-1)^n)/3)
%o A003683 (PARI) a(n)=(2^n-(-1)^n)<<(n-1)/3 \\ _Charles R Greathouse IV_, Apr 17 2012
%o A003683 (Sage) [lucas_number1(n,2,-8) for n in range(0, 24)] # _Zerinvary Lajos_, Apr 22 2009
%o A003683 (Magma) [2^(n-1)*(2^n - (-1)^n)/3: n in [0..30]]; // _Vincenzo Librandi_, Aug 19 2011
%Y A003683 Cf. A001045, A003674, A011782, A054881, A192232, A192382, A238801.
%K A003683 nonn,easy
%O A003683 0,3
%A A003683 _N. J. A. Sloane_
%E A003683 Erroneous references to spanning trees in K_2 X P_n deleted by Frans Faase, Feb 07 2009