cp's OEIS Frontend

A094832 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) = 4.

%I A094832 #27 Jul 02 2021 16:46:12
%S A094832 1,3,10,34,117,406,1417,4965,17443,61390,216318,762841,2691574,
%T A094832 9500193,33539833,118428835,418214706,1476968554,5216307805,
%U A094832 18423344550,65070265609,229827800509,811757757123,2867166603766,10127007608998
%N A094832 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) = 4.
%C A094832 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.
%H A094832 Michael De Vlieger, <a href="/A094832/b094832.txt">Table of n, a(n) for n = 0..1824</a>
%H A094832 Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.
%H A094832 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9,1).
%F A094832 a(n+1) = 3*a(n) + A094833(n-1). - _Philippe Deléham_, Mar 18 2007
%F A094832 a(n) = (2/9)*Sum_{r=1..8} sin(r*Pi/3)*sin(4*r*Pi/9)*(2*cos(r*Pi/9))^(2n).
%F A094832 a(n) = 6a(n-1) - 9a(n-2) + a(n-3).
%F A094832 G.f.: (-1+3x-x^2)/(-1+6x-9x^2+x^3).
%F A094832 a(n) = A094833(n+2) - 3*A094833(n+1). - _Philippe Deléham_, Mar 18 2007
%t A094832 LinearRecurrence[{6,-9,1},{1,3,10},30] (* _Harvey P. Dale_, May 18 2011 *)
%Y A094832 Cf. A094833.
%K A094832 nonn
%O A094832 0,2
%A A094832 _Herbert Kociemba_, Jun 13 2004