0,2

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

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

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

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

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

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

a(n)=sum{j=0..1, sum{k=0..n, C(2n+j, 2k) }} - Paul Barry (pbarry(AT)wit.ie), Nov 29 2003

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

Row sums of triangle A134316. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 19 2007

a(n)=A011782(n)*A003945(n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 08 2009]

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

with(finance):seq(ceil(futurevalue(3, 3, n)), n=-1..21); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 25 2009]

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]

(MAGMA) [ (3*4^n+0^n)/4: n in [0..22] ]; [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Aug 15 2009]

First difference of 4^n (A000302).

Cf. A134316.

Sequence in context: A151167 A064562 A077828 this_sequence A164346 A113956 A103943

Adjacent sequences: A001998 A001999 A002000 this_sequence A002002 A002003 A002004

nonn,easy,new

N. J. A. Sloane (njas(AT)research.att.com).