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 A004054 Expansion of (1-x)/((1+x)*(1-2*x)*(1-3*x)). 3
 1, 3, 11, 35, 111, 343, 1051, 3195, 9671, 29183, 87891, 264355, 794431, 2386023, 7163531, 21501515, 64526391, 193622863, 580955971, 1743042675, 5229477551, 15689131703, 47068793211, 141209175835 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of paths with n+2 steps on the cycle graph C_6 which start at the first node and end at the 3rd node and each step is -1, 0 or +1. - Herbert Kociemba, Sep 30 2020 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 X. Acloque, Polynexus Numbers and other mathematical wonders. Index entries for linear recurrences with constant coefficients, signature (4,-1,-6). FORMULA From Paul Barry, Sep 13 2003: (Start) The sequence 0, 0, 1, ... has a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*A001045(2*k). a(n) = 3^n/6 + (-1)^n/6 - 0^n/6 - 2^n/6. (End) From Xavier Acloque, Oct 17 2003: (Start) a(n) = 3^n - 2^n - (-1^(n-1)). a(n) = A001047(n) - (-1^(n-1)). (End) The signed sequence 0, 1, -3, ... has g.f. x*(1+x)/((1-x)*(1+2*x)*(1+3*x)) and a(n) = 1/6 + (-2)^n/3 - (-3)^n/2. It is the third inverse binomial transform of A001045(2*n-1) - 0^n/2. - Paul Barry, Apr 21 2004 From Paul Barry, Jul 22 2004: (Start) Convolution of A000244 and A078008. a(n) = Sum_{k=0..n} A078008(k)*3^(n-k). a(n) = (3*A000244(n) - A001045(n+2))/2. (End) MATHEMATICA Table[1/6 ((-1)^(2+n)-2^(n+2)+3^(n+2)), {n, 0, 30}] (* Herbert Kociemba, Sep 30 2020 *) PROG (Magma) [Ceiling(3^(n+2)/6+(-1)^(n+2)/6-0^n/6-2^(n+2)/6) : n in [0..30]]; // Vincenzo Librandi, Oct 08 2011 (PARI) Vec((1-x)/((1+x)*(1-2*x)*(1-3*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012 CROSSREFS Cf. A001045, A001047. Cf. A000244, A078008. Sequence in context: A026125 A026154 A025181 * A068995 A109196 A032637 Adjacent sequences: A004051 A004052 A004053 * A004055 A004056 A004057 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified March 27 09:01 EDT 2023. Contains 361555 sequences. (Running on oeis4.)