Search: id:A002001
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%I A002001
%S A002001 1,3,12,48,192,768,3072,12288,49152,196608,786432,3145728,
%T A002001 12582912,50331648,201326592,805306368,3221225472,12884901888,
%U A002001 51539607552,206158430208,824633720832,3298534883328,13194139533312
%N A002001 a(n) = 3*4^(n-1), n>0; a(0)=1.
%C A002001 Second binomial transform of (1,1,4,4,16,16,...)=(3*2^n+(-2)^n)/4. - 
               Paul Barry (pbarry(AT)wit.ie), Jul 16 2003
%C A002001 Number of vertices (or sides) formed after the (n-1)-th iterate towards 
               building a Koch's snowflake. - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Jan 24 2005
%C A002001 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 R. Janjic 
               (agnus(AT)blic.net), Mar 27 2007
%H A002001 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas 
               for Some Functions on Finite Sets</a>
%H A002001 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=456">
               Encyclopedia of Combinatorial Structures 456</a>
%H A002001 R. J. Krawcyk, <a href="http://www.iit.edu/~krawczyk/koch.html">Koch 
               Curve</a>
%H A002001 C. Lanius, <a href="http://math.rice.edu/~lanius/frac/koch/koch.html">
               The Koch Snowflake</a>
%H A002001 P. Kernan, <a href="http://theory2.phys.cwru.edu/~pete/java_chaos/KochApplet.html">
               Koch Snowflake</a>
%F A002001 a(n)=(3*4^n+0^n)/4 (with 0^0=1). E.g.f. (3exp(4x)+1)/4. - Paul Barry 
               (pbarry(AT)wit.ie), Apr 20 2003
%F A002001 With interpolated zeros, this has e.g.f. (3*cosh(2x)+1)/4 and binomial 
               transform A006342. - Paul Barry (pbarry(AT)wit.ie), Sep 03 2003
%F A002001 a(n)=sum{j=0..1, sum{k=0..n, C(2n+j, 2k) }} - Paul Barry (pbarry(AT)wit.ie), 
               Nov 29 2003
%F A002001 G.f.: (1-x)/(1-4x). The sequence 1, 3, -12, 48, -192... has g.f. (1+7x)/
               (1+4x) - Paul Barry (pbarry(AT)wit.ie), Feb 12 2004
%F A002001 Row sums of triangle A134316. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Oct 19 2007
%F A002001 a(n)=A011782(n)*A003945(n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Jul 08 2009]
%p A002001 with(combstruct):ZL0:=S=Prod(Sequence(Prod(a, Sequence(b))), a):ZL1:=Prod(begin_blockP, 
               Z, end_blockP):ZL2:=Prod(begin_blockLR, Z, Sequence(Prod(mu_length, 
               Z), card>=1), end_blockLR): ZL3:=Prod(begin_blockRL, Sequence(Prod(mu_length, 
               Z), card>=1), Z, end_blockRL):Q:=subs([a=Union(ZL1,ZL1,ZL1), b=ZL1], 
               ZL0), begin_blockP=Epsilon, end_blockP=Epsilon, begin_blockLR=Epsilon, 
               end_blockLR=Epsilon, begin_blockRL=Epsilon, end_blockRL=Epsilon, 
               mu_length=Epsilon:temp15:=draw([S, {Q}, unlabelled], size=15):seq(count([S, 
               {Q}, unlabelled], size=n)/3, n=1..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Mar 08 2008
%p A002001 with(finance):seq(ceil(futurevalue(3,3,n)), n=-1..21);# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Mar 25 2009]
%t A002001 a=1;s=a;lst={a};Do[AppendTo[lst,a=3*s];s=a+s,{n,5!}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Nov 10 2009]
%o A002001 (MAGMA) [ (3*4^n+0^n)/4: n in [0..22] ]; [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), 
               Aug 15 2009]
%Y A002001 First difference of 4^n (A000302).
%Y A002001 Cf. A134316.
%Y A002001 Sequence in context: A151167 A064562 A077828 this_sequence A164346 A113956 
               A103943
%Y A002001 Adjacent sequences: A001998 A001999 A002000 this_sequence A002002 A002003 
               A002004
%K A002001 nonn,easy,new
%O A002001 0,2
%A A002001 N. J. A. Sloane (njas(AT)research.att.com).

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