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%I #141 Sep 26 2023 06:19:53
%S 1,3,12,48,192,768,3072,12288,49152,196608,786432,3145728,12582912,
%T 50331648,201326592,805306368,3221225472,12884901888,51539607552,
%U 206158430208,824633720832,3298534883328,13194139533312,52776558133248,211106232532992,844424930131968
%N a(n) = 3*4^(n-1), n>0; a(0)=1.
%C Second binomial transform of (1,1,4,4,16,16,...) = (3*2^n+(-2)^n)/4. - _Paul Barry_, Jul 16 2003
%C Number of vertices (or sides) formed after the (n-1)-th iterate towards building a Koch's snowflake. - _Lekraj Beedassy_, Jan 24 2005
%C For n >= 1, a(n) is the number of functions f:{1,2,...,n}->{1,2,3,4} such that for a fixed x in {1,2,...,n} and a fixed y in {1,2,3,4} we have f(x) <> y. - Aleksandar M. Janjic and _Milan Janjic_, Mar 27 2007
%C a(n) = (n+1) terms in the sequence (1, 2, 3, 3, 3, ...) dot (n+1) terms in the sequence (1, 1, 3, 12, 48, ...). Example: a(4) = 192 = (1, 2, 3, 3, 3) dot (1, 1, 3, 12, 48) = (1 + 2 + 9 + 36 + 144). - _Gary W. Adamson_, Aug 03 2010
%C a(n) is the number of compositions of n when there are 3 types of each natural number. - _Milan Janjic_, Aug 13 2010
%C See A178789 for the number of acute (= exterior) angles of the Koch snowflake referred to in the above comment by L. Beedassy. - _M. F. Hasler_, Dec 17 2013
%C After 1, subsequence of A033428. - _Vincenzo Librandi_, May 26 2014
%C a(n) counts walks (closed) on the graph G(1-vertex; 1-loop x3, 2-loop x3, 3-loop x3, 4-loop x3, ...). - _David Neil McGrath_, Jan 01 2015
%C For n > 1, a(n) are numbers k such that (2^(k-1) mod k)/(2^k mod k) = 2; 2^(a(n)-1) == 2^(2n-1) (mod a(n)) and 2^a(n) == 2^(2n-2) (mod a(n)). - _Thomas Ordowski_, Apr 22 2020
%C For n > 1, a(n) is the number of 4-colorings of the Hex graph of size 2 X (n-1). More generally, for q > 2, the number of q-colorings of the Hex graph of size 2 X n is given by q*(q - 1)*(q - 2)^(2*n - 2). - _Sela Fried_, Sep 25 2023
%H Vincenzo Librandi, <a href="/A002001/b002001.txt">Table of n, a(n) for n = 0..300</a>
%H Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, <a href="https://arxiv.org/abs/2203.13205">Honeycombs in the Pascal triangle and beyond</a>, arXiv:2203.13205 [math.HO], 2022. See p. 5.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=456">Encyclopedia of Combinatorial Structures 456</a>.
%H P. Kernan, <a href="http://theory2.phys.cwru.edu/~pete/java_chaos/KochApplet.html">Koch Snowflake</a>. [Broken link]
%H C. Lanius, <a href="http://math.rice.edu/~lanius/frac/koch/koch.html">The Koch Snowflake</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KochSnowflake.html">Koch Snowflake</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Koch_snowflake">Koch snowflake</a>.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>.
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (4).
%F From _Paul Barry_, Apr 20 2003: (Start)
%F a(n) = (3*4^n + 0^n)/4 (with 0^0=1).
%F E.g.f.: (3*exp(4*x) + 1)/4. (End)
%F With interpolated zeros, this has e.g.f. (3*cosh(2*x) + 1)/4 and binomial transform A006342. - _Paul Barry_, Sep 03 2003
%F a(n) = Sum_{j=0..1} Sum_{k=0..n} C(2n+j, 2k). - _Paul Barry_, Nov 29 2003
%F G.f.: (1-x)/(1-4*x). The sequence 1, 3, -12, 48, -192, ... has g.f. (1+7*x)/(1+4*x). - _Paul Barry_, Feb 12 2004
%F a(n) = 3*Sum_{k=0..n-1} a(k). - _Adi Dani_, Jun 24 2011
%F G.f.: 1/(1-3*Sum_{k>=1} x^k). - _Joerg Arndt_, Jun 24 2011
%F Row sums of triangle A134316. - _Gary W. Adamson_, Oct 19 2007
%F a(n) = A011782(n) * A003945(n). - _R. J. Mathar_, Jul 08 2009
%F If p(1)=3 and p(i)=3 for i > 1, and if A is the Hessenberg matrix of order n defined by A(i,j) = p(j-i+1) when i <= j, A(i,j)=-1 when i=j+1, and A(i,j) = 0 otherwise, then, for n >= 1, a(n) = det A. - _Milan Janjic_, Apr 29 2010
%F a(n) = 4*a(n-1), a(0)=1, a(1)=3. - _Vincenzo Librandi_, Dec 31 2010
%F G.f.: 1 - G(0) where G(k) = 1 - 1/(1-3*x)/(1-x/(x-1/G(k+1))); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Jan 25 2013
%F G.f.: x+2*x/(G(0)-2), where G(k) = 1 + 1/(1 - x*(3*k+1)/(x*(3*k+4) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 26 2013
%F a(n) = ceiling(3*4^(n-1)). - _Wesley Ivan Hurt_, Dec 17 2013
%F Construct the power matrix T(n,j) = [A(n)^*j]*[S(n)^*(j-1)] where A(n)=(3,3,3,...) and S(n)=(0,1,0,0,...). (* is convolution operation.) Then T(n,j) counts n-walks containing j loops on the single vertex graph above and a(n) = Sum_{j=1..n} T(n,j). (S(n)^*0=I.) - _David Neil McGrath_, Jan 01 2015
%p A002001:=n->ceil(3*4^(n-1)); seq(A002001(n), n=0..30); # _Wesley Ivan Hurt_, Dec 17 2013
%t Table[Ceiling[3*4^(n - 1)], {n, 0, 30}] (* _Wesley Ivan Hurt_, May 26 2014 *)
%o (Magma) [ (3*4^n+0^n)/4: n in [0..22] ]; // _Klaus Brockhaus_, Aug 15 2009
%o (PARI) v=vector(100,n,3*4^(n-2));v[1]=1;v \\ _Charles R Greathouse IV_, May 19 2011
%o (PARI) A002001=n->if(n,3*4^(n-1),1) \\ _M. F. Hasler_, Dec 17 2013
%Y First difference of 4^n (A000302).
%Y Cf. A003945, A006342, A011782, A033428, A134316, A178789.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Dec 11 1996
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