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A002001
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a(n) = 3*4^(n-1), n>0; a(0)=1.
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42
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1, 3, 12, 48, 192, 768, 3072, 12288, 49152, 196608, 786432, 3145728, 12582912, 50331648, 201326592, 805306368, 3221225472, 12884901888, 51539607552, 206158430208, 824633720832, 3298534883328, 13194139533312, 52776558133248, 211106232532992, 844424930131968
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OFFSET
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0,2
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COMMENTS
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Second binomial transform of (1,1,4,4,16,16,...) = (3*2^n+(-2)^n)/4. - Paul Barry, Jul 16 2003
Number of vertices (or sides) formed after the (n-1)-th iterate towards building a Koch's snowflake. - Lekraj Beedassy, Jan 24 2005
For n>=1, a(n) is equal to 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
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
a(n) is the number of compositions of n when there are 3 types of each natural number. - Milan Janjic, Aug 13 2010
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
After 1, subsequence of A033428. - Vincenzo Librandi, May 26 2014
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
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..300
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 456
R. J. Krawcyk, Koch Curve
C. Lanius, The Koch Snowflake
P. Kernan, Koch Snowflake
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (4).
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FORMULA
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a(n) = (3*4^n+0^n)/4 (with 0^0=1). E.g.f.: (3*exp(4*x)+1)/4. - Paul Barry, Apr 20 2003
With interpolated zeros, this has e.g.f. (3*cosh(2*x)+1)/4 and binomial transform A006342. - Paul Barry, Sep 03 2003
a(n) = sum{j=0..1, sum{k=0..n, C(2n+j, 2k) }}. - Paul Barry, Nov 29 2003
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
a(n) = 3*sum(k=0..n-1, a(k)). - Adi Dani, Jun 24 2011
G.f.: 1/(1-3*sum(k>=1,x^k)). - Joerg Arndt, Jun 24 2011
Row sums of triangle A134316. - Gary W. Adamson, Oct 19 2007
a(n) = A011782(n) * A003945(n). - R. J. Mathar, Jul 08 2009
If p[1]=3, p[i]=3, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. - Milan Janjic, Apr 29 2010
a(n) = 4*a(n-1), a(0)=1, a(1)=3. - Vincenzo Librandi, Dec 31 2010
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
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
a(n) = ceiling(3*4^(n-1)). - Wesley Ivan Hurt, Dec 17 2013
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
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MAPLE
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A002001:=n->ceil(3*4^(n-1)); seq(A002001(n), n=0..30); # Wesley Ivan Hurt, Dec 17 2013
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MATHEMATICA
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a=1; s=a; lst={a}; Do[AppendTo[lst, a=3*s]; s=a+s, {n, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 10 2009 *)
Table[Ceiling[3*4^(n - 1)], {n, 0, 30}] (* Wesley Ivan Hurt, May 26 2014 *)
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PROG
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(MAGMA) [ (3*4^n+0^n)/4: n in [0..22] ]; // Klaus Brockhaus, Aug 15 2009
(PARI) v=vector(100, n, 3*4^(n-2)); v[1]=1; v \\ Charles R Greathouse IV, May 19, 2011
(PARI) A002001=n->if(n, 3*4^(n-1), 1) \\ M. F. Hasler, Dec 17 2013
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CROSSREFS
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First difference of 4^n (A000302).
Cf. A134316.
Sequence in context: A254942 A077828 A164346 * A113956 A103943 A283679
Adjacent sequences: A001998 A001999 A002000 * A002002 A002003 A002004
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, Dec 11 1996
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STATUS
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approved
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