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A003946
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Expansion of (1+x)/(1-3*x).
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127
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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
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OFFSET
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0,2
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COMMENTS
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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 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
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
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LINKS
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FORMULA
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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
(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)
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EXAMPLE
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G.f. = 1 + 4*x + 12*x^2 + 36*x^3 + 108*x^4 + 324*x^5 + 972*x^6 + 2916*x^7 + ...
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MAPLE
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if n = 0 then 1 else 4*3^(n-1); fi;
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MATHEMATICA
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CoefficientList[Series[(1 + x)/(1 - 3 x), {x, 0, 30}], x] (* Vincenzo Librandi, Dev 11 2012 *)
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PROG
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(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)$
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CROSSREFS
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KEYWORD
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nonn,easy,nice,walk
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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