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A094834 Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = 3, s(2n+1) = 6.

%I A094834 #24 Feb 12 2022 17:48:27
%S A094834 1,5,21,82,308,1131,4096,14705,52497,186733,662630,2347680,8309143,
%T A094834 29388368,103895601,367187437,1297452581,4583924154,16193659132,
%U A094834 57204089987,202065531888,713750040577,2521114546457,8905002445437
%N A094834 Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = 3, s(2n+1) = 6.
%C A094834 In general, a(n) = (2/m)*Sum_{r=1..m-1} sin(r*j*Pi/m)*sin(r*k*Pi/m)*(2*cos(r*Pi/m))^(2n+1) counts (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < m and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = j, s(2n+1) = k.
%C A094834 This sequence is the odd bisection of A188048. - _John Blythe Dobson_, Jun 20 2015
%H A094834 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9,1).
%F A094834 a(n) = (2/9)*Sum_{r=1..8} sin(r*Pi/3)*sin(2*r*Pi/3)*(2*cos(r*Pi/9))^(2n+1).
%F A094834 a(n) = 6*a(n-1) - 9*a(n-2) + a(n-3).
%F A094834 G.f.: x(-1+x)/(-1 + 6x - 9x^2 + x^3).
%F A094834 a(n) = A094829(n+1) - A094829(n). - _R. J. Mathar_, Nov 15 2019
%t A094834 CoefficientList[Series[(x - 1)/(- 1 + 6 x - 9 x^2 + x^3), {x, 0, 20}], x] (* _Vincenzo Librandi_, Jun 21 2015 *)
%t A094834 LinearRecurrence[{6,-9,1},{1,5,21},30] (* _Harvey P. Dale_, Dec 27 2019 *)
%o A094834 (Magma) I:=[1,5,21]; [n le 3 select I[n] else 6*Self(n-1)-9*Self(n-2)+Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Jun 21 2015
%K A094834 nonn,easy
%O A094834 1,2
%A A094834 _Herbert Kociemba_, Jun 13 2004