A002001 a(n) = 3*4^(n-1), n>0; a(0)=1.

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

Offset: 0

Author

N. J. A. Sloane, Dec 11 1996

Keywords

• nonn
• easy

Comments

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 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

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

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

For n > 1, a(n) is the number of pixels in the HEALPix discretization of the sphere of order n-2; HEALPix is a common sphere pixellization scheme in astronomy, cosmology, and nuclear engineering. - Jayson R. Vavrek, Aug 08 2024

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, Honeycombs in the Pascal triangle and beyond, arXiv:2203.13205 [math.HO], 2022. See p. 5.
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 456.
• P. Kernan, Koch Snowflake. [Broken link]
• C. Lanius, The Koch Snowflake.
• Eric Weisstein's World of Mathematics, Koch Snowflake.
• Wikipedia, Koch snowflake.
• Index to divisibility sequences.
• Index entries for linear recurrences with constant coefficients, signature (4).

Crossrefs

First difference of 4^n (A000302).

Cf. A003945, A006342, A011782, A033428, A134316, A178789.

Programs

Magma

[ (3*4^n+0^n)/4: n in [0..22] ]; // Klaus Brockhaus, Aug 15 2009

Maple

A002001:=n->ceil(3*4^(n-1)); seq(A002001(n), n=0..30); # Wesley Ivan Hurt, Dec 17 2013

Mathematica

Table[Ceiling[3*4^(n - 1)], {n, 0, 30}] (* Wesley Ivan Hurt, May 26 2014 *)

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

Formula

From Paul Barry, Apr 20 2003: (Start)

a(n) = (3*4^n + 0^n)/4 (with 0^0=1).

E.g.f.: (3*exp(4*x) + 1)/4. (End)

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 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

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