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 A003946 Expansion of (1+x)/(1-3*x). 126
 1, 4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236196, 708588, 2125764, 6377292, 19131876, 57395628, 172186884, 516560652, 1549681956, 4649045868, 13947137604, 41841412812, 125524238436, 376572715308, 1129718145924 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Coordination sequence for infinite tree with valency 4. The n-th term of the coordination sequence of the infinite tree with valency 2m is the same as the number of reduced words of size n in the free group on m generators. In the five sequences A003946, A003948, A003950, A003952, A003954 m is 2, 3, 4, 5, 6. - Avi Peretz (njk(AT)netvision.net.il), Feb 23 2001 and Ola Veshta (olaveshta(AT)my-deja.com), Mar 30 2001 a(n) is the number of nonreversing random walks of the length of n edges on a two-dimensional square lattice, all beginning at a fixed point P. - Pawel P. Mazur (Pawel.Mazur(AT)pwr.wroc.pl), Apr 06 2005 Binomial transform of {1, 3, 5, 11, 21, 43, ...}, see A001045. Binomial transform is {1, 5, 21, 85, 341, 1365, ...}, see A002450. - Philippe Deléham, Jul 22 2005 For n >= 2, a(n) is equal to the number of functions f:{1,2,...,n+1}->{1,2,3} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,3} we have f(x_1) <> y_1 and f(x_2) <> y_2. - Milan Janjic, Apr 19 2007 Equals row sums of triangle A143865. - Gary W. Adamson, Sep 04 2008 Equals INVERT transform of the odd integers = 1/(1 - x - 3x^2 - 5x^3 - ...). - Gary W. Adamson, Jul 27 2009 a(n) is the number of generalized compositions of n+1 when there are 2 *i-1 different types of the part i, (i=1,2,...). - Milan Janjic, Aug 26 2010 Number of length-n strings of 4 letters with no two adjacent letters identical. The general case (strings of r letters) is the sequence with g.f. (1+x)/(1-(r-1)*x). - Joerg Arndt, Oct 11 2012 The sequence is the INVERTi transform of A015448: (1, 5, 21, 89, 377, ...). - Gary W. Adamson, Aug 06 2016 Let D(m) = {d(m,i)}, i = 1..q, denote the set of the q divisors of a number m, and consider s1(m) and s2(m) the sums of the divisors that are congruent to 1 and 2 (mod 3) respectively. For n > 0, the sequence a(n) lists the numbers m such that s1(m) = 5 and s2(m) = 2. - Michel Lagneau, Feb 09 2017 a(n) is the number of quaternary sequences of length n such that no two consecutive terms have distance 2. - David Nacin, May 31 2017 Also the number of maximal cliques in the n-Sierpinski sieve graph. - Eric W. Weisstein, Dec 01 2017 Number of 3-permutations of n elements avoiding the patterns 231, 321. See Bonichon and Sun. - Michel Marcus, Aug 19 2022 LINKS T. D. Noe, Table of n, a(n) for n = 0..200 Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, (an + b)-color compositions, arXiv:1707.07798 [math.CO], 2017. Nicolas Bonichon and Pierre-Jean Morel, Baxter d-permutations and other pattern avoiding classes, arXiv:2202.12677 [math.CO], 2022. D. J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996. John Elias, Illustration: Sierpinski Hexagrams I. M. Gessel and Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 305 Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets A. M. Nemirovsky et al., Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers, J. Statist. Phys., 67 (1992), 1083-1108. Nathan Sun, On d-permutations and Pattern Avoidance Classes, arXiv:2208.08506 [math.CO], 2022. Eric Weisstein's World of Mathematics, Maximal Clique Eric Weisstein's World of Mathematics, Sierpinski Sieve Graph Index entries for linear recurrences with constant coefficients, signature (3). FORMULA a(n) = floor(4*3^(n-1)). - Michael Somos, Jun 18 2002 a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 2. - Philippe Deléham, Jul 10 2005 The Hankel transform of this sequence is [1,-4,0,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 21 2007 a(n + 1) = (((1 + sqrt(-11))/2)^n + ((1 - sqrt(-11))/2)^n)^2 - (((1 + sqrt(-11))/2)^n - ((1 - sqrt(-11))/2)^n)^2. - Raphie Frank, Dec 07 2015 From Mario C. Enriquez, Apr 01 2017: (Start) (L(a(n+k)) - 1)/a(n) reduces to the form C/a(n-1), where n > 1, k >= 0, L(a(n)) is the a(n)-th Lucas number and C = (L(a(n+k)) - 1)/3. (L(a(n+k)) - 1)/3 mod (L(a(n)) - 1)/3 = (L(a(n)) - 1)/3 - 1, where n >= 1, k >= 0 and L(a(n)) is the a(n)-th Lucas number. (End) EXAMPLE G.f. = 1 + 4*x + 12*x^2 + 36*x^3 + 108*x^4 + 324*x^5 + 972*x^6 + 2916*x^7 + ... MAPLE if n = 0 then 1 else 4*3^(n-1); fi; MATHEMATICA Join[{1}, 4 3^Range[0, 30]] (* Vladimir Joseph Stephan Orlovsky, Jun 14 2009 *) Join[{1}, NestList[3# &, 4, 30]] (* Harvey P. Dale, Nov 30 2011 *) CoefficientList[Series[(1 + x)/(1 - 3 x), {x, 0, 30}], x] (* Vincenzo Librandi, Dev 11 2012 *) Join[{1}, LinearRecurrence[{3}, {4}, 20]] (* Eric W. Weisstein, Dec 01 2017 *) PROG (PARI) {a(n) = if( n<1, n==0, 4 * 3^(n-1))}; /* Michael Somos, Jun 18 2002 */ (PARI) Vec((1+x)/(1-3*x) + O(x^100)) \\ Altug Alkan, Dec 07 2015 (Maxima) A003946[n]:=if n<1 then 1 else 4*3^(n-1)\$ makelist(A003946[n], n, 0, 30); /* Martin Ettl, Oct 29 2012 */ (Magma)  cat [4*3^(n-1): n in [1..25]]; // Vincenzo Librandi, Dec 11 2012 CROSSREFS Cf. A029653, A143865, column 4 in A265583, A015448. Sequence in context: A168969 A169017 A169065 * A052156 A169113 A169161 Adjacent sequences:  A003943 A003944 A003945 * A003947 A003948 A003949 KEYWORD nonn,easy,nice,walk AUTHOR EXTENSIONS Additional comments from Michael Somos, Jun 18 2002 Edited by N. J. A. Sloane, Dec 04 2009 STATUS approved

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Last modified October 6 21:32 EDT 2022. Contains 357270 sequences. (Running on oeis4.)