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Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 3.
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%I #28 Feb 13 2022 02:37:16

%S 1,2,6,19,62,207,703,2417,8382,29242,102431,359790,1266103,4460939,

%T 15730497,55500634,195890270,691566411,2441886670,8623112591,

%U 30453261927,107553444913,379864424726,1341658806066,4738726458775

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

%C 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) counts (s(0), s(1), ..., s(2n)) such that 0 < s(i) < m and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = j, s(2n) = k.

%C A comparison of their recurrence relations shows that this sequence is the even bisection of A188048. - _John Blythe Dobson_, Jun 20 2015

%H Michael De Vlieger, <a href="/A094831/b094831.txt">Table of n, a(n) for n = 0..1825</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9,1)

%F a(n) = (2/9) * Sum_{r=1..8} sin(r*Pi/3)^2*(2*cos(r*Pi/9))^(2*n).

%F a(n) = 6*a(n-1) - 9*a(n-2) + a(n-3).

%F G.f.: (1-4*x+3*x^2)/(1-6*x+9*x^2-x^3).

%t CoefficientList[Series[(1 - 4 x + 3 x^2)/(1 - 6 x + 9 x^2 - x^3), {x, 0, 24}], x] (* _Michael De Vlieger_, Feb 12 2022 *)

%o (PARI) Vec((1-4*x+3*x^2)/(1-6*x+9*x^2-x^3) + O(x^30)) \\ _Michel Marcus_, Jun 21 2015

%Y Cf. A188048.

%K nonn

%O 0,2

%A _Herbert Kociemba_, Jun 13 2004