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 A002001 a(n) = 3*4^(n-1), n>0; a(0)=1. 47
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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 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). 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 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 *) PROG (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 CROSSREFS First difference of 4^n (A000302). Cf. A003945, A006342, A011782, A033428, A134316, A178789. Sequence in context: A254942 A077828 A164346 * A113956 A323261 A103943 Adjacent sequences: A001998 A001999 A002000 * A002002 A002003 A002004 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Dec 11 1996 STATUS approved

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Last modified March 3 19:35 EST 2024. Contains 370512 sequences. (Running on oeis4.)